![CHAPTER I
Random Walk and Brownian
Motion
1 WHAT IS A STOCHASTIC PROCESS?
Denoting by X„ the value of a stock at an nth unit of time, one may represent
its (erratic) evolution by a family of random variables {X0 , X,, ...} indexed by
the discrete-time parameter n E 7L + . The number X, of car accidents in a city
during the time interval [0, t] gives rise to a collection of random variables
{ X1 : t >, 0} indexed by the continuous-time parameter t. The velocity X. at a
point u in a turbulent wind field provides a family of random variables
{X: u e l83 } indexed by a multidimensional spatial parameter u. More generally
we make the following definition.
Definition 1.1. Given an index set I, a stochastic process indexed by I is a
collection of random variables {X1 : 2 e I} on a probability space (Cl, ., P)
taking values in a set S. The set S is called the state space of the process.
In the above, one may take, respectively: (i) I = Z , S = I!; (ii) I = [0, oo),
S = Z; (iii) I = l, S = X83 . For the most part we shall study stochastic
processes indexed by a one-dimensional set of real numbers (e.g., time). Here
the natural ordering of numbers coincides with the sense of evolution of the
process. This order is lost for stochastic processes indexed by a multidimensional
parameter; such processes are usually referred to as random fields. The state
space S will often be a set of real numbers, finite, countable, (i.e., discrete) or
uncountable. However, we also allow for the possibility of vector-valued
variables. As a matter of convenience in notation the index set is often suppressed
when the context makes it clear. In particular, we often write {X„} in place of
{X„: n = 0, 1, 2, ...} and {X,} in place of {X,: t >, 0}.
For a stochastic process the values of the random variables corresponding](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-20-320.jpg)
![TRANSIENCE AND RECURRENCE PROPERTIES OF THE SIMPLE RANDOM WALK 5
3 TRANSIENCE AND RECURRENCE PROPERTIES OF THE
SIMPLE RANDOM WALK
Let us first consider the manner in which a particle escapes from an interval.
Let TY denote the first time that the process starting at x reaches y, i.e.
Ty:= min{n >, 0:S„ = y}. (3.1)
To avoid trivialities, assume 0 <p < 1. For integers c and d with c < d, denote
4(x):= P(T < T' ). (3.2)
In other words, 4(x) is the probability that the particle starting at x reaches d
before it reaches c. Since in one step the particle moves to x + I with probability
p, or to x — 1 with probability q, one has
4(x) = po(x + 1) + q4(x — 1) (3.3)
so that
O(x+ 1)—O(x)=-[O(x)—cß(x— 1)], c+ 1 ,<x,<d— 1
p
0(c) = 0,
(3.4)
q(d) = 1.
Thus, ¢(x) is the solution to the discrete boundary-value problem (3.4). For
p ^ q, Eq. 3.4 yields
x-1 x-1 q Y
O(x) = Z [^(y + 1 ) — o(y)] = Z - [O(c + 1) — O(c)]
v=c v=c P
x
-'
=0(c+1) Y - =0(c+1)
1
1-
- (q/P)
/p)(3.5)
yc ' P)
To determine 4(c + 1) take x = d in Eq. 3.5 to get
1 =4(d)=4(c+ 1) 1 — (qlP)°-`
1 — q/P
Then
1 — q/P
q(c + 1) = 1 — (glp)d c](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-24-320.jpg)
![6 RANDOM WALK AND BROWNIAN MOTION
so that
P(Tx<Tx)= 1 —(q/P)x for c<x<d, p q. (3.6)
1 — (q/P)d-c
Now let
0/i(x)==P(T, < Tdx). (3.7)
By symmetry (or the same method as above),
P(Tx<Td)= 1—(P/q)d-xfor c<x<d,p q. (3.8)
1 — (P/q)
d-c
Note that O(x) + fr(x) = 1, proving that the particle starting in the interior of
[c, d] will eventually reach the boundary (i.e., either c or d) with probability
1. Now if c < x, then (Exercise 3)
P({S„} will ever reach c) = P(T,' < oo) = lim i(x)
d-+oo
x
—
•
um 9/
ifp 1
>2
= dam°°—
P
c
1, ifp <Z,
x—c
q if p>
= P(3.9)
1, ifp<Z.
By symmetry, or as above,
i
P({S:} will ever reach d) = P(Td' < oo) = 1' d_x f p > 2 (3.10)
C
i
q/ ,
f p < Z .
Observe that one gets from these calculations the (geometric) distribution
function for the extremes Mx = sup,, S„ and mx = inf„ S; (Exercise 7).
Note that, by the strong law of large numbers (Chapter 0),
x
P S„ = x+S„
^p—gasn--•oo =1. (3.11)
n n](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-25-320.jpg)
![8 RANDOM WALK AND BROWNIAN MOTION
Similarly, whatever the initial position x < d,
P({S.} will eventually reach d) = P(Td < oo)
= lim x—c = 1. (3.17)
e -- ao d — c
Thus, no matter where the particle may be initially, it will eventually reach any
given state y with probability 1. After having reached y for the first time, it will
move to y + 1 or to y — 1. From either of these positions the particle is again
bound to reach y with probability 1, and so on. In other words (Exercise 4),
P(S' = y i.o.) = 1, if p = 9 = 2. (3.18)
This argument is discussed again in Example 4.1 of Chapter II.
Definition 3.2. A state y for which Eq. 3.18 holds is called recurrent. If all
states are recurrent, then the stochastic process is called a recurrent process.
Let r^X denote the time of the first return to x,
rl„ := inf{n >, 1: S.' = x} . (3.19)
Then, conditioning on the first step, it will follow (Exercise 6) that
P(f],, < oo) = 2 min(p, q). (3.20)
4 FIRST PASSAGE TIMES FOR THE SIMPLE RANDOM WALK
Consider the random variable 7 := T° representing the first time the simple
random walk starting at zero reaches the level (state) y. We will calculate the
distribution of T
y by means of an analysis of the sample paths of the simple
random walk. Let FN,y = {Ty = N} denote the event that the particle reaches
state y for the first time at the Nth step. Then,
FN.y ={S„iky for n=0,1,...,N-1,SN =y}. (4.1)
Note that "SN = y" means that there are (N + y)/2 plus l's and (N — y)/2
minus 1's among XI , X2 , ... , XN (see Eq. 2.1). Therefore, we assume that
IYI <, N and N + y is even. Now there are as many paths leading from (0, 0)
to (N, y) as there are ways of choosing (N + y)/2 plus l's among X1 , X2 , ... , XN ,
namely
N
N+y
2](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-27-320.jpg)
![12 RANDOM WALK AND BROWNIAN MOTION
The corresponding position process is defined by
Sö=x, S"= x+X 1 +•••+Xn , ni1. (5.2)
The case k = 1 is that already treated in the preceding sections with p = q = 2.
In particular, for k = 1 we know that the simple symmetric random walk is
recurrent.
Consider the coordinates of X. = (X,.. . , X.). Although X,', and X„ are not
independent, notice that they are uncorrelated for i 0 j. Likewise, it follows that
the coordinates of the position vector S = (Sn' 1 , ... , S) are uncorrelated.
In particular,
ES„ = x,
xi xj if =j
Cov(Sn ' , Sn')
= tn, i (5.3)
0, ifi*j.
Therefore the covariance matrix of S. is nI where I is the k x k identity matrix.
The problem of describing the recurrence properties of the simple symmetric
random walk in k dimensions is solved by the following theorem of Pö1ya.
Theorem 5.1. [P61ya]. {S„} is recurrent for k = 1, 2 and transient for k ? 3.
Proof. The result has already been obtained for k = 1. In general, let S. = Sn
o
and write
rn=P(Sn=0)
fn = P(S" = 0 for the first time after time 0 at n), n >, 1. (5.4)
Then we get the convolution equation
n
rn = fj r"_jforn=1,2,...,
j=0
ro=1, f0= 0 . (5.5)
Let P(s) and f(s) denote the respective probability generating functions of {rn}
and {f,.} defined by
P(s) _ f (s) _ > fn s" (0 < s < 1). (5.6)
n =o n =o
The convolution equation (5.5) transforms as
P(s) = 1 + I E .ijrn-js'sn-j = 1 + Z
(MW
=0
Y rm sm)f
j sj = 1 + f(s)f(s). (5.7)
n=1j =0 j =0](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-31-320.jpg)
![14 RANDOM WALK AND BROWNIAN MOTION
that
1 (2n)
= z" (P;.m)
2
(5.12)
2 n
is nearly an average of pj,m's (with respect to the pj,m distribution). In any case,
22n
(2n)
jmax Pj,m]Pj,m= 2a" ( n
n) ma
x Pj.m. (5.13)
j,m j,m
The maximum value of pj,m is attained at j and m nearest to n/3 (Exercise 5).
Therefore, writing [x] for the integer part of x,
i
r2n 1( 2n)
1 n.
(5.14)
22n n 3" rn i fl, [n],
Apply Stirling's formula to get (see 5.19 below),
r 2n -
C
2" nn n n
3/2 for some C' > 0. (5.15)
In particular,
Er"<oo. (5.16)
"
The general case, r2n < ck n-k/2 for k > 3, is left as an exercise (Exercise 1).
n
The constants appearing in the estimate (5.15) are easily computed from the
monotonicity of the ratio n!/{(2nn)`I2n"e- "}; whose limit as n -> oo is 1
according to Stirling's formula. To see that the ratio is monotonically decreasing,
simply observe that
t
logn!= log n! — flog n — n log n + n — log(2n) li 2
(2nn)112n"e- "
J.j log j— Z log n}—{n logn—n}—log(2n)"2
,-1 )
= j log(j — 1) + log(j) — f " log x dx } + 1 — log(2n)l iz
(
U2 2 J^ J
(5.17)](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-33-320.jpg)
![18 RANDOM WALK AND BROWNIAN MOTION
p=2+ 2^ o and 0= ^
Here p and a are two fixed numbers, or > 0. Then as f --> cc, the mean
displacement t f (p — q)0 converges to tµ and the variance converges to tae . In
the limit, then, the position X, of the particle at time t > 0 is Gaussian with
probability density function (in y) given by
_ z
P(t; x, Y) _ (2ita2t)1j2 eXp{—
(Y
_
2QZt tµ) (7.1)
Ifs > 0 then X, + — X, is the sum of displacements during the time interval
(t, t + s]. Therefore, by the argument above, X, +s — X, is Gaussian with mean
sµ and variance sae, and it is independent of {X,,: 0 < u < t}. In particular, for
every finite set of time points 0 < tl < t2 < • • • <t, the random variables
X,,, X^2 — X,.. . , X XX,,,_, are independent. A stochastic process with this
last property is said to be a process with independent increments. This is the
continuous-time analogue of random walks. From the physical description of
the process {X} as representing (a coordinate of) the path of a diffusing solute
particle, one would expect that the sample paths of the process (i.e., the
trajectories t —* X(w) = w,) may be taken to be continuous. That this is indeed
the case is an important mathematical result originally due to Norbert Wiener.
For this reason, Brownian motion is also called the Wiener process. A complete
definition of Brownian motion goes as follows.
Definition 7.1. A Brownian motion with drift µ and diffusion coefficient a2 is
a stochastic process {X,: t 0} having continuous sample paths and
independent Gaussian increments with mean and variance of an increment
XX+s — XX being sp and sae , respectively. If X0 = x, then this Brownian motion
is said to start at x. A Brownian motion with zero drift and diffusion coefficient
of 1 is called the standard Brownian motion.
Families of random variables {X} constituting Brownian motions arise in
many different contexts on diverse probability spaces. The canonical model for
Brownian motion is given as follows.
1. The sample space S2 := C[0, oo) is the set of all real-valued continuous
functions on the time interval [0, cc). This is the set of all possible
trajectories (sample paths) of the process.
2. XX (co) := co, is the value of the sample path w at time t.
3. S2 is equipped with the smallest sigmafield .y of subsets of S2
containing the class .moo of all finite-dimensional sets of the form
F = fce e ): a; <w,. < bi , i = 1, 2, ... , k}, where a; <b, are constants
and 0 < t l < t2 < • • • < tk are a finite set of time points. .F is said to be
generated by .moo.](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-37-320.jpg)
, (8.2)
V "
where [nt] is the integer part of nt. Figure 8.1 plots the sample path of
{X;"^: t >, 0} up to time t = 13/n if the successive displacements take values
Z1 =-1, Zz =+1, Z3 =+1, Z4 =+1, Z5 =-1, Z6 =+1, Z7 =+1,
Zg = — 1, Zq = + 1, Z10 = + 1, 211 = + 1, Z12 = — 1.
_ 1
Simi
.—i
‚In
4 —4 '-4
Vn
3
^---; •-4
'In
? —4—.--1-4
Vn
Vn
1 1 3 4 5 6 7 i 9 10 11 12 13 t
n n n n n n It n n n n n n
Intl
EX' ") = 0, VarX^11 = n ~ t,
Cov(Vt., + `v l)= [
n] ~ S.
Figure 8.1](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-39-320.jpg)
![THE FUNCTIONAL CENTRAL LIMIT THEOREM (FCLT) 21
The process {S11 : t >, 0} records the discrete-time random walk
{S.: m = 0, 1, 2, ...} on a continuous time scale whose unit is n times that of
the discrete time unit, i.e., Sm is plotted at time m/n. The process
{X} = {(1//)S[„f] } also scales distance by measuring distances on a scale
whose unit is f times the unit of measurement used for the random walk.
This is a convenient normalization, since
EX ) = 0, Var X}") =
z
[nt]0 2
tazfor large n. (8.3)
n
In a time interval (t 1 , t2] the overall "displacement" X — X(°) is the sum of
a large number [ntz] — [nt,] n(t2 — t,) of small i.i.d. random variables
1 1
In the case {Z,„} is i.i.d. Bernoulli, this means reducing the step sizes of the
random variables to t1 = 1/,.fn. In a physical application, looking at {X}
means
means the following.
1. The random walk is observed at times t, < t2 <t3 < • • • sufficiently far
apart to allow a large number of individual displacements to occur during
each of the time intervals (t,, tz], (tz, t3], ... , and
2. Measurements of distance are made on a "macroscopic" scale whose unit
of measurement is much larger than the average magnitude of the
individual displacements. The normalizing large parameter n scales time
and n'' z scales space coordinates.
Since the sample paths of {X} have jumps (though small for large n) and
are, therefore, discontinuous, it is technically more convenient to linearly
interpolate the random walk between one jump point and the next, using the
same space—time scales as used for {X°}. The polygonal process {X,(" 1 } is
formally defined by
Xt") = SIntl + (nt — [nt])t 0. (8.4)
In this way, just as for the limiting Brownian motion process, the paths of
{X} are continuous. Figure 8.2 plots the path of {X1"°} corresponding to the
path of {X} drawn in Figure 8.1. In a time interval m/n < t < (m + 1)/n, X;")
is constant at level 1// S„„ while X}") changes linearly from l/ f S. at time
t=m/n to
I S„, +1 = S
"' Z'" + ' at time
m + 1
me t =
n](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-40-320.jpg)
![22 RANDOM WALK AND BROWNIAN MOTION
I [n(] Z101j+j
S^ rl + (t —
n ) ^n
Vn
4
Vn
3
‚In
2
do
W,
= 0, VarX^rn) _
[nt] + 1 (t — [nt] )2
t
n n n
[ns]
Figure 8.2
Thus, in any given interval [0, T] the maximum difference between the two
processes {X,(n»} and {X,(°) } does not exceed
c n (T) = max IZII , IZ21 , IZ[ nT,+1I
To see that the difference between {X,(n) } and {X;n) } is negligible for large n,
consider the following estimate. For each 6 > 0,
P(en (T) > 6) = 1 — P(en (T) < (5)
= I —P( IZ^'<(5 for allm= 1,2,...,[nT]+ 1)
l V n
= 1 — (P(IZ11 < 5))
[nT1+1
= 1 — (1 — P(IZ11 > 6.^ n))[nT1 +l (8.5)
Assuming for simplicity that EIZ1 13 < co, Chebyshev's inequality yields
P(1Z11 > 6^) <, EIZII 3/63n3/2 . Use this in (8.5) to get (Exercise 9)
EIZIr
(nTJ+l
P(e (T) > (5) 1 — ( i — (533/2 )
1—exp{— EIZ113T }—► 0 (8.6)
63n1/2
when n is large. Here indicates that the difference between the two sides](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-41-320.jpg)
![THE FUNCTIONAL CENTRAL LIMIT THEOREM (FCLT) 23
goes to zero. Thus, on any closed and bounded time interval the behaviors of
{X,(") } and {X} are the same in the large-n limit.
Note that given any finite set of time points 0 < t 1 < t2 < < t, the joint
distribution of (X, X;z) , .. . , X(") ) converges to the finite-dimensional
distribution of (XX1 , X,2 , ... , Xtk ), where {X,} is a Brownian motion with zero
drift and diffusion coefficient a 2 . To see this, note that X, X — Xt"^, ... ,
X,(k ) — X() , are independent random variables that by the classical central limit
theorem (Chapter 0) converge in distribution to Gaussian random variables
with zero means and variances t1a2 , (t2 — t, )a 2 , ... , (tk — tk_ t )Q 2 . That is to
say, the joint distribution of (X, X,(") — X°, X (") — X ("^ )converges to
that of (X,,, X X, ... , X X,k _,). By a linear transformation, one gets
the desired convergence of finite-dimensional distributions of {X(" ) } (and,
therefore, of {X^"1 }) to those of the Brownian motion process {X} (Exercise 1).
Roughly speaking, to establish the full convergence in distribution of {X!" 1}
to Brownian motion, one further looks at a finite set of time points comprising
a fine subdivision of a bounded interval [0, T] and shows that the fluctuations
of the process {X^"^} on [0, T] between successive points of this subdivision
are sufficiently small in probability, a property called the tightness of the process.
This control over fluctuations together with the convergence of {X^"1} evaluated
at the time points of the subdivision ensures convergence in distribution to a
continuous process whose finite-dimensional distributions are the same as those
of Brownian motion (see theoretical complements for details). Since there is no
process other than Brownian motion with continuous sample paths that has
these limiting finite-dimensional distributions, it follows that the limit must be
Brownian motion.
A precise statement of the functional central limit theorem (FCLT) is the
following.
Theorem 8.1. (The Functional Central Limit Theorem). Suppose {Z,,,:
m = 1, 2, ...} is an i.i.d. sequence with EZ„, = 0 and variance a 2 > 0. Then as
n —* cc the stochastic processes {X: t > 0} (or {Xr"°: t >, 0}) converge in
distribution to a Brownian motion starting at the origin with zero drift and
diffusion coefficient a 2 .
An important way in which to view the convergence asserted in the FCLT
is as follows. First, the sample paths of the polygonal process {X^" 1 } belong to
the Space S = C[O, oo) of all continuous real-valued function on [0, oo), as do
those of the limiting Brownian motion {X}. This space C[O, oo) is a metric
space with a natural notion of convergence of sequences {d")}, say, being that
"{m(")} converges to w in C[O, co) as n tends to infinity if and only if
{co(") (t): a <, t < b} converges uniformly to {w(t): a < t < b} for all closed and
bounded intervals [a, b]." Second, the distributions of the processes {X} and
{X,} are probability measures P" and P on a certain class F of events of C[0, cc),
called Borel subsets, which is generated by and therefore includes all of the
finite-dimensional events. .F includes as well various important infinite-](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-42-320.jpg)
![RECURRENCE PROBABILITIES FOR BROWNIAN MOTION 25
complement 2)
c—x d—xl
P(i < rd) = lim P( {i} reaches ------- before ----/)
"- xQ 6
= lim P({S,„} reaches c" before d"), (9.4)
"-+00
where
c"= Lc -x ;],
6
and
d—
x n if d" = d —X is an integer,
d =
" d—x
d x+ 1 if not.
By relation (3.14) of Section 3, one has
d_ x -
- n
a
P(rx <t) = l
d
im " = lim ----- -- . (9.5)
Therefore,
P(r, <ra) = d—c(c<x<d,µ=0). (9.6)
Similarly, using relation (3.13) of Section 3 instead of (3.14), one gets
P(ta <T') = x_c(c <x<d,p =0). (9.7)
Letting d --* + oo in (9.6) and c —+ — co in (9.7), one has
P(rc < oo) = P({X, } ever reaches c) = I (c < x, p = 0),
(9.8)
P(r< < oo) = P({X; } ever reaches d) = 1 (x < d, p = 0).
The relations (9.8) mean that a Brownian motion with zero drift is recurrent,](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-44-320.jpg)
![26 RANDOM WALK AND BROWNIAN MOTION
just as a simple symmetric random walk was shown to be in Section 3.
The next problem is to calculate the corresponding probabilities when the
drift is a nonzero quantity p. Consider, for each large n, the Bernoulli sequence
P(Z,n.n=+1)=Pn=1+
p
2 26^ ,
{Z,„ n : m = 1, 2, ...} with
1
P(Zm,n= —1)=9n =---------.
µ
22
Write Sm,n =Zl ,n +— , +Z,„ n for m>,1,So.,,=0.Then,
[nt] µ
EX (n) = ES1n,1,n = a n tp
^nor(9.9)
Var X^ [
nt] Var Z„n — —
[nt] (
(1 1 µ )Z) t,
(n) = 1
n n 7
and a slight modification of the FCLT, with no significant difference in proof,
implies that {X;n)} and, therefore, {X;n°} converges in distribution to a Brownian
motion with drift µ/Q and diffusion coefficient of I that starts at the origin. Let
{X'} be a Brownian motion with drift It and diffusion coefficient a 2 starting at
x. Then {W = (X, — x)/Q} is a Brownian motion with drift p/a and diffusion
coefficient of! that starts at the origin. Hence, by using relation (3.8) of Section 3,
P(i, <x) = P({X, } reaches c before d)
= Pl{W} reaches c — x before
d — x^
Q a
= lim ({Sm,n : m = 0, 1, 2, ...} reaches cn before dn )
n-• ro
d=x
1 — (ihn /^]n) a f
= lim / d-x - c=x
1 — (pn/qn) a
✓n a ✓n
µ d=x
1 + a n
Q.J
1-
1— 7
= um
n-ml^ d—c J
1 + a n
U n
1— I I
— ; µ^](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-45-320.jpg)
![28 RANDOM WALK AND BROWNIAN MOTION
distribution
N
P(7.=N)=IYI N+y 1
N
Y
2
N=IYI>IYI+2,IYI+4..... (10.1)
Now let r = T° be the first time a standard Brownian motion starting at the
origin reaches z. Let {X^") } be the polygonal process corresponding to the simple
symmetric random walk. Considering the first time {X ) } reaches z, one has
by the FLCT (Theorem 8.1) and Eq. 10.1 (Exercise 1),
P(a= > t) = lim P(TZ f] > [nt])
n- X
= lim P(T=,n] = N)
n-+m N=(nt]+1
N
= lim
IYI
( N+
y N
(Y = [z^])
n-+ao N=tnt]+1, N 2
N—yeven
(10.2)
Now according to Stirling's formula, for large integers M, we can write
M! = (21r) 2e -MMnr+2 (1 + SM ) (10.3)
where 8M —► 0 as M —► oo. Since y = [z], N> [nt], and 2(N ± y) >
{[nt] — I[z/]I}/2, both N and N + y tend to infinity as n —• oo. Therefore,
for N + y even,
Ne-NNN+#2-N
IYI N + y 2 _ N =IYI 2
N2 (2ir) t N e -(N+Y)12(N + Y)
(N+y)/2+Ie
—(N—Y)/2 (N — Y
l (N-Yu2+#
2 ` 2 J
X (1 + o(1))
(2ir)1I2N312 1+ N I 1— N (1 + o( 1 ))
I (N + Y)/ 2 (N — Y)l2
(2ir)
2
/2N3/2 (1 + ) 1 — (1 + o( 1 )),
N
(10.4)
where o(1) denotes a quantity whose magnitude is bounded above by a quantity
en (t, z) that depends only on n, t, z and which goes to zero as n —• oo. Also,](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-47-320.jpg)
![FIRST PASSAGE TIME DISTRIBUTIONS FOR BROWNIAN MOTION 29
r y (N+yuz y wN-vuz _ N + Y Y _ YzIYI3
log
[
(1
+ N) 1 - N 2 N 2N2 +ß(N3)
+N 2 y [
N+2N +O INI3/]
2 3
_ -2N+8(N,y), (10.5)
where IB(N, y)j < n -11 z c(t, z) and c(t, z) is a constant depending only on t and
z. Combining (10.4) and (10.5), we have
N
(
^NN + Y 2-N =nN3I/2exp
1-
z
2N}(1 + o( 1 ))
2
(
= n I N312 exp1-2N}(1 + 0(1)),
(10.6)
where o(1) --, 0 as n -* oo, uniformly for N> [nt], N - [z] even. Using this
in (10.2), one obtains
( z
P(r= > t) = lim rz IN312 expj
-2N}.
(10.7)
n—^ao N> n:J, l
N—[z.n) even
Now either [nt] + 1 or [nt] + 2 is a possible value of N. In the first case the
sum in (10.7) is over values N = [nt] - I + 2r (r = 1, 2, ...), and in the second
case over values [nt] + 2r (r = 1, 2, ...). Since the differences in corresponding
values of N/n are 2/n, one may take N = [nt] + 2r for the purpose of calculating
(10.7). Thus,
P(t2 > t) = lim ] + 2ex ( nz2
}
fl X) n ([nt] + 2r)31^ p 1 2([nt] + 2r)
_-Izl lim ^ 1
21 exp^-
n r1 2 nzz 1
-.. = (t + 2r/n)3122(t + 2r/n)
_ j2r
fl
- z2^
Izl 2 u3/^ exp - 2u du. (10.8)
Now, by the change of variables v = Izi/ f , we get
2
folz
P(T= > t) _ v e - "Z/z dv. (10.9)
^](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-48-320.jpg)
![30 RANDOM WALK AND BROWNIAN MOTION
The first passage time distribution for the more general case of Brownian
motion {X1} with zero drift and diffusion coefficient Qz > 0, starting at the
origin, is now obtained by applying (10.9) to the standard Brownian motion
{(1/Q)X}. Therefore,
2 fI=Ibf
P(;> t)_ e-°2/2 dv. (10.10)
o
The probability density function fQ2(t) of; is obtained from (10.10) as
f 2(t) = Izle-ZZna=t (t > 0). (10.11)
(2nci2 )1/2 t3/2
Note that for large t the tail of the p.d.f. f 2 (t) is of the order of t -3/2 . Therefore,
although {X°} will reach z in a finite time with probability 1, the expected time
is infinite (Exercise 11).
Consider now a Brownian motion {X,} with a nonzero drift µ and diffusion
coefficient a2 that starts at the origin. As in Section 9, the polygonal process
{X^n)} corresponding to the simple random walk S,„,n = Z1 ,ß + • • • + Z„,,n,
S0,, = 0, with P(Ztn,n = 1) = p„ = 2 + µ/(2Q), converges in distribution to
{W = Xja}, which is a Brownian motion with drift µ/u and diffusion coefficient
1. On the other hand, writing T
y,n for the first passage time of {S„, n : m = 0, 1, ...}
to y, one has, by relation (4.6) of Section 4,
N
N
P(1 = IY) p(N+v)/2R(N-v)/2
N) = N N + y
2
N
IYI N ) (N-i-y)12( l
(N-Y)l2
N+y 2 - 1+
2—
N Ql Q n
FYI µ2 N/21 µ y/2 p y/2
= N N + y 12_N(
a2
1 n 1 + /
l —
2U n ; )
(10.12)
For y = [w..J] for some given nonzero w, and N = [nt] + 2r for some given
t> 0 and all positive integers r, one has
( _
N/2 /
I1 1 } ^
Jv/z/l — ^
l-y/2
/ µz +r J µ Wf/z µ w,,/„/z
a2n
)nt/2
1 + a nI l Q^)
(1 + 0(1))](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-49-320.jpg)
![FIRST PASSAGE TIME DISTRIBUTIONS FOR BROWNIAN MOTION 31
= ex
2 2 r
tµex ^w ex 1
l + 0
exp{ _ 4}( i — a2
n^ p 26 p 2v (
())
t/1 2 /LW
µz)n]rin
(=exp--2+6
-zn1 +o(1))
r
i ( z l ro
exp — t
µ
2 + ^W exp —Y-
2 + e„
J (l + 0(1)) (10.13)
2a Cr l Q
where E„ does not depend on r and goes to zero as n —+ oc, and o(l) represents
a term that goes to zero uniformly for all r >, 1 as n --* x. The first passage
time i2 to z for {X1 } is the same as the first passage time to w = z/a for the
process {I4'} = {X/a}. It follows from (9.12), (9.13) that if
(i) p. <0 and z > O or
(ii) p > 0 and z < 0,
then there is a positive probability that the process { W} will never reach w = z/a
(i.e., tZ = co). On the other hand, the sum of the probabilities in (10.12) over
N> [nt] only gives the probability that the random walk reaches [w,,/n
-] in a
finite time greater than [nt]. By the FCLT, (10.6)—(10.8) and (10.12) and (10.13),
we have
2
P(t < r < co) = IwI exp — tu2 + w
µ lim 1
z
n 2a Q „^^ r _ 1 n(t + 2r/n)312
wz
x exp —
+
+ e )'^”
2(t 2r/n)
^ tµ2 jJ
11 f w2
_ ^
(wI exp — 2a2 + 2 V312 exp — 2v
x [
expf—^Zn`„-`I/2
du
= I I p
^ —t1iW/.1
1
(2n)i/z w ex 2a
a
v3/z
W2
----
2
x exp{— — (v — t) dv
2v 2az
z z
= w^ exp
^
1 W/I)
1 exp — W — v A,
(2n)l/Z
a fj ^
v3'22v 2U2
for w = z/a. (10.14)](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-50-320.jpg)
a"
= 2(2nn 1
)2/ 2n =
( 2n)(^)'"
n
= P(S2" = 0),
where we have adopted the convention that (2 2
n 1 ) = 0 in writing the middle
equality.](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-52-320.jpg)
![THE BROWNIAN BRIDGE 35
where I'(2n1 is defined in Theorem 11.1. By Theorem 11.1 and Stirling's
approximation
lim P(I'(Z"1 < 2nx) = lim [^1
(2k)!(2n — 2k)!
2_2n
n-w n-•. k=o (k!)2 ((n — k)!)z
1-1 (27r) 112e - 2k(2k)zk+ j
= lim Y-
112 -k k+})2
n-.00 k=o ((2n) e k
(2x)1/2e-2(n-k)(2(n — k))[2cn-k)+#12-2n
x ((2n)'1'e-("-k)(n — k)n-k++)2
[nxxl
= lim Y-
n-+oo k=o 7i (n — k)
1/2
1 Intl 11_ 1x 1
= n li k= n k k 112 n o (y(1 — y)) 1I2 dy
•
n^ 1
n//
The following (invariance) corollary follows by applying the FCLT.
Corollary 11.3. Let {Z,, Z2, ...} be a sequence of i.i.d. random variables such
that EZ, = 0, EZi = 1. Then, defining {X^"1} as in (8.4) and y(n1 as above, one has
lim P(y (n) x) = 2 sin - ' ^. (11.7)
n-^M n
From the arcsine law of the time of the last visit to zero it is also possible
to get the distribution of the length of time in [0, 1] the standard Brownian
motion spends on the positive side of the origin (i.e., an occupation time
law) again.as an arcsine distribution. This fact is recorded in the following
corollary (Exercise 2).
Corollary 11.4. Let U = I {t < 1: Bt e IT + }I, where I I denotes Lebesgue
measure and {B1} is standard Brownian motion starting at 0. Then,
P(U < x) = 2 sin -1 ‚/i(11.8)
n
12 THE BROWNIAN BRIDGE
Let {B,} be a standard Brownian motion starting at zero. Since B, — tBl vanishes
for t = 0 and t = 1, the stochastic process {B*} defined by](https://image.slidesharecdn.com/1221535-250606213134-e6efc192/85/Stochastic-Processes-With-Applications-Classics-In-Applied-Mathematics-61-Rabi-N-Bhattacharya-54-320.jpg)
Stochastic Processes With Applications Classics In Applied Mathematics 61 Rabi N Bhattacharya
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- 7. Books in the Classics in Applied Mathematics series are monographs and textbooks declared out of print by their original publishers, though they are of continued importance and interest to the mathematical community. SIAM publishes this series to ensure that the information presented in these texts is not lost to today's students and researchers. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board John Boyd, University of Michigan Leah Edelstein-Keshet, University of British Columbia William G. Faris, University of Arizona Nicholas J. Higham, University of Manchester Peter Hoff, University of Washington Mark Kot, University of Washington Peter Olver, University of Minnesota Philip Protter, Cornell University Gerhard Wanner, L'Universite de Geneve Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The Symmetric Eigenvalue Problem Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Ba§ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes *First time in print.
- 8. Classics in Applied Mathematics (continued) Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kale and Malcolm Slaney, Principles of Computerized Tomographic Imaging R. Wong, Asymptotic Approximations of Integrals O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed-Point Theorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimization and Economic Theory Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. I: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods Leah Edelstein-Keshet, Mathematical Models in Biology Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations J. L. Hodges, Jr. and E. L. Lehmann, Basic Concepts of Probability and Statistics, Second Edition George F. Carrier, Max Krook, and Carl E. Pearson, Functions of a Complex Variable: Theory and Technique Friedrich Pukelsheim, Optimal Design of Experiments Israel Gohberg, Peter Lancaster, and Leiba Rodman, Invariant Subspaces of Matrices with Applications Lee A. Segel with G. H. Handelman, Mathematics Applied to Continuum Mechanics Rajendra Bhatia, Perturbation Bounds for Matrix Eigenvalues Barry C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics Charles A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties Stephen L. Campbell and Carl D. Meyer, Generalized Inverses of Linear Transformations Alexander Morgan, Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials Galen R. Shorack and Jon A. Wellner, Empirical Processes with Applications to Statistics Richard W. Cottle, Jong-Shi Pang, and Richard E. Stone, The Linear Complementarity Problem Rabi N. Bhattacharya and Edward C. Waymire, Stochastic Processes with Applications Robert J. Adler, The Geometry of Random Fields Mordecai Avriel, Walter E. Diewert, Siegfried Schaible, and Israel Zang, Generalized Concavity Rabi N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions
- 9. F ^ Stochastic Processes with Applications b ci Rabi N. Bhattacharya University of Arizona Tucson, Arizona Edward C. Waymire Oregon State University Corvallis, Oregon pia m o Society for Industrial and Applied Mathematics Philadelphia
- 10. Copyright © 2009 by the Society for Industrial and Applied Mathematics This SIAM edition is an unabridged republication of the work first published by John Wiley & Sons (SEA) Pte. Ltd., 1992. 10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Library of Congress Cataloging-in-Publication Data Bhattacharya, R. N. (Rabindra Nath), 1937- Stochastic processes with applications / Rabi N. Bhattacharya, Edward C. Waymire. p. cm. -- (Classics in applied mathematics ; 61) Originally published: New York : Wiley, 1990. Includes index. ISBN 978-0-898716-89-4 1. Stochastic processes. I. Waymire, Edward C. II. Title. QA274.B49 2009 519.2'3--dc22 2009022943 S1L2JTL. is a registered trademark.
- 11. To Gouri and Linda, with love
- 12. Contents Preface to the Classics Edition xiii Preface xv Sample Course Outline xvii I Random Walk and Brownian Motion 1 1. What is a Stochastic Process?, 1 2. The Simple Random Walk, 3 3. Transience and Recurrence Properties of the Simple Random Walk, 5 4. First Passage Times for the Simple Random Walk, 8 5. Multidimensional Random Walks, 11 6. Canonical Construction of Stochastic Processes, 15 7. Brownian Motion, 17 8. The Functional Central Limit Theorem (FCLT), 20 9. Recurrence Probabilities for Brownian Motion, 24 10. First Passage Time Distributions for Brownian Motion, 27 11. The Arcsine Law, 32 12. The Brownian Bridge, 35 13. Stopping Times and Martingales, 39 14. Chapter Application: Fluctuations of Random Walks with Slow Trends and the Hurst Phenomenon, 53 Exercises, 62 Theoretical Complements, 90 II Discrete-Parameter Markov Chains 109 1. Markov Dependence, 109 2. Transition Probabilities and the Probability Space, 110 ix
- 13. X CONTENTS 3. Some Examples, 113 4. Stopping Times and the Strong Markov Property, 117 5. A Classification of States of a Markov Chain, 120 6. Convergence to Steady State for Irreducible and Aperiodic Markov Processes on Finite Spaces, 126 7. Steady-State Distributions for General Finite-State Markov Processes, 132 8. Markov Chains: Transience and Recurrence Properties, 135 9. The Law of Large Numbers and Invariant Distributions for Markov Chains, 138 10. The Central Limit Theorem for Markov Chains, 148 11. Absorption Probabilities, 151 12. One-Dimensional Nearest-Neighbor Gibbs States, 162 13. A Markovian Approach to Linear Time Series Models, 166 14. Markov Processes Generated by Iterations of I.I.D. Maps, 174 15. Chapter Application: Data Compression and Entropy, 184 Exercises, 189 Theoretical Complements, 214 III Birth—Death Markov Chains 233 1. Introduction to Birth—Death Chains, 233 2. Transience and Recurrence Properties, 234 3. Invariant Distributions for Birth—Death Chains, 238 4. Calculations of Transition Probabilities by Spectral Methods, 241 5. Chapter Application: The Ehrenfest Model of Heat Exchange, 246 Exercises, 252 Theoretical Complements, 256 IV Continuous-Parameter Markov Chains 261 1. Introduction to Continuous-Time Markov Chains, 261 2. Kolmogorov's Backward and Forward Equations, 263 3. Solutions to Kolmogorov's Equations in Exponential Form, 267 4. Solutions to Kolmogorov's Equations by Successive Approximation, 271 5. Sample Path Analysis and the Strong Markov Property, 275 6. The Minimal Process and Explosion, 288 7. Some Examples, 292 8. Asymptotic Behavior of Continuous-Time Markov Chains, 303 9. Calculation of Transition Probabilities by Spectral Methods, 314 10. Absorption Probabilities, 318
- 14. CONTENTS Xi 11. Chapter Application: An Interacting System: The Simple Symmetric Voter Model, 324 Exercises, 333 Theoretical Complements, 349 V Brownian Motion and Diffusions 367 1. Introduction and Definition, 367 2. Kolmogorov's Backward and Forward Equations, Martingales, 371 3. Transformation of the Generator under Relabeling of the State Space, 381 4. Diffusions as Limits of Birth—Death Chains, 386 5. Transition Probabilities from the Kolmogorov Equations: Examples, 389 6. Diffusions with Reflecting Boundaries, 393 7. Diffusions with Absorbing Boundaries, 402 8. Calculation of Transition Probabilities by Spectral Methods, 408 9. Transience and Recurrence of Diffusions, 414 10. Null and Positive Recurrence of Diffusions, 420 11. Stopping Times and the Strong Markov Property, 423 12. Invariant Distributions and the Strong Law of Large Numbers, 432 13. The Central Limit Theorem for Diffusions, 438 14. Introduction to Multidimensional Brownian Motion and Diffusions, 441 15. Multidimensional Diffusions under Absorbing Boundary Conditions and Criteria for Transience and Recurrence, 448 16. Reflecting Boundary Conditions for Multidimensional Diffusions, 460 17. Chapter Application: G. I. Taylor's Theory of Solute Transport in a Capillary, 468 Exercises, 475 Theoretical Complements, 497 VI Dynamic Programming and Stochastic Optimization 519 1. Finite-Horizon Optimization, 519 2. The Infinite-Horizon Problem, 525 3. Optimal Control of Diffusions, 533 4. Optimal Stopping and the Secretary Problem, 542 5. Chapter Application: Optimality of (S, s) Policies in Inventory Problems, 549 Exercises, 557 Theoretical Complements, 559
- 15. xii CONTENTS VII An Introduction to Stochastic Differential Equations 563 1. The Stochastic Integral, 563 2. Construction of Diffusions as Solutions of Stochastic Differential Equations, 571 3. It6's Lemma, 582 4. Chapter Application: Asymptotics of Singular Diffusions, 591 Exercises, 598 Theoretical Complements, 607 0 A Probability and Measure Theory Overview 625 1. Probability Spaces, 625 2. Random Variables and Integration, 627 3. Limits and Integration, 631 4. Product Measures and Independence, Radon—Nikodym Theorem and Conditional Probability, 636 5. Convergence in Distribution in Finite Dimensions, 643 6. Classical Laws of Large Numbers, 646 7. Classical Central Limit Theorems, 649 8. Fourier Series and the Fourier Transform, 653 Author Index 665 Subject Index 667 Errata 673
- 16. Preface to the Classics Edition The publication of Stochastic Processes with Applications (SPWA) in the SIAM Classic in Applied Mathematics series is a matter of great pleasure for us, and we are deeply appreciative of the efforts and good will that went into it. The book has been out of print for nearly ten years. During this period we received a number of requests from instructors for permission to make copies of the book to be used as a text on stochastic processes for graduate students. We also received many kind laudatory words, along with inquiries about the possibility of bringing out a second edition, from mathematicians, statisticians, physicists, chemists, geoscientists, and others from the U.S. and abroad. We hope that the inclusion of a detailed errata is a helpful addition to the original. SPWA was a work of love for its authors. As stated in the original preface, the book was intended for use (1) as a graduate-level text for students in diverse disciplines with a reasonable background in probability and analysis, and (2) as a reference on stochastic processes for applied mathematicians, scientists, engineers, economists, and others whose work involves the application of probability. It was our desire to communicate our sense of excitement for the subject of stochastic processes to a broad community of students and researchers. Although we have often empha- sized substance over form, the presentation is systematic and rigorous. A few proofs are relegated to Theoretical Complements, and appropriate references for proofs are provided for some additional advanced technical material. The book covers a sub- stantial part of what we considered to be the core of the subject, especially from the point of view of applications. Nearly two decades have passed since the publication of SPWA, but the importance of the subject has only grown. We are very happy to see that the book's rather unique style of exposition has a place in the broader applied mathematics literature. We would like to take this opportunity to express our gratitude to all those col- leagues who over the years have provided us with encouragement and generous words on this book. Special thanks are due to SIAM editors Bill Faris and Sara Murphy for shepherding SPWA back to print. RABI N. BHATTACHARYA EDWARD C. WAYMIRE XIII
- 17. Preface This is a text on stochastic processes for graduate students in science and engineering, including mathematics and statistics. It has become somewhat commonplace to find growing numbers of students from outside of mathematics enrolled along with mathematics students in our graduate courses on stochastic processes. In this book we seek to address such a mixed audience. For this purpose, in the main body of the text the theory is developed at a relatively simple technical level with some emphasis on computation and examples. Sometimes to make a mathematical argument complete, certain of the more technical explanations are relegated to the end of the chapter under the label theoretical complements. This approach also allows some flexibility in instruction. A few sample course outlines have been provided to illustrate the possibilities for designing various types of courses based on this book. The theoretical complements also contain some supplementary results and references to the literature. Measure theory is used sparingly and with explanation. The instructor may exercise control over its emphasis and use depending on the background of the majority of the students in the class. Chapter 0 at the end of the book may be used as a short course in measure theoretical probability for self study. In any case we suggest that students unfamiliar with measure theory read over the first few sections of the chapter early on in the course and look up standard results there from time to time, as they are referred in the text. Chapter applications, appearing at the end of the chapters, are largely drawn from physics, computer science, economics, and engineering. There are many additional examples and applications illustrating the theory; they appear in the text and among the exercises. Some of the more advanced or difficult exercises are marked by asterisks. Many appear with hints. Some exercises are provided to complete an argument or statement in the text. Occasionally certain well-known results are only a few steps away from the theory developed in the text. Such results are often cited in the exercises, along with an outline of steps, which can be used to complete their derivation. Rules of cross-reference in the book are as follows. Theorem m.n, Proposition xv
- 18. xvi PREFACE m.n, or Corollary m.n, refers to the nth such assertion in section m of the same chapter. Exercise n, or Example n, refers to the nth Exercise, or nth Example, of the same section. Exercise m.n (Example m.n) refers to Exercise n (Example n) of a different section m within the same chapter. When referring to a result or an example in a different chapter, the chapter number is always mentioned along with the label m.n to locate it within that chapter. This book took a long time to write. We gratefully acknowledge research support from the National Science Foundation and the Army Research Office during this period. Special thanks are due to Wiley editors Beatrice Shube and Kate Roach for their encouragement and assistance in seeing this effort through. RABI N. BHATTACHARYA EDWARD C. WAYMIRE Bloomington, Indiana Corvallis, Oregon February 1990
- 19. Sample Course Outlines COURSE I Beginning with the Simple Random Walk, this course leads through Brownian Motion and Diffusion. It also contains an introduction to discrete/continuous- parameter Markov Chains and Martingales. More emphasis is placed on concepts, principles, computations, and examples than on complete proofs and technical details. Chapter 1 Chapter II Chapter III §1-7 (+ Informal Review of Chapter 0, §4) §1-4 §1--3 §13 (Up to Proposition 13.5) §5 (By examples) §5 §11 (Example 2) §13 Chapter IV Chapter V Chapter VI §1-7 (Quick survey §1 §4 by examples) §2 (Give transience/recurrence from Proposition 2.5) §3 (Informal justification of equation (3.4) only) §5-7 §10 §11 (Omit proof of Theorem 11.1) §12-14 COURSE 2 The principal topics are the Functional Central Limit Theorem, Martingales, Diffusions, and Stochastic Differential Equations. To complete proofs and for supplementary material, the theoretical complements are an essential part of this course. Chapter I Chapter V Chapter VI Chapter VII §1-4 (Quick survey) §1-3 §4 §1--4 §6-10 §6-7 §13 §11 §13-17 COURSE 3 This is a course on Markov Chains that also contains an introduction to Martingales. Theoretical complements may he used only sparingly. Chapter I Chapter II Chapter III Chapter IV Chapter VI §1-6 §1-9 §1 §1-11 §1-2 §13 §11 §5 §4-5 §12 or 15 §13-14 xvii
- 20. CHAPTER I Random Walk and Brownian Motion 1 WHAT IS A STOCHASTIC PROCESS? Denoting by X„ the value of a stock at an nth unit of time, one may represent its (erratic) evolution by a family of random variables {X0 , X,, ...} indexed by the discrete-time parameter n E 7L + . The number X, of car accidents in a city during the time interval [0, t] gives rise to a collection of random variables { X1 : t >, 0} indexed by the continuous-time parameter t. The velocity X. at a point u in a turbulent wind field provides a family of random variables {X: u e l83 } indexed by a multidimensional spatial parameter u. More generally we make the following definition. Definition 1.1. Given an index set I, a stochastic process indexed by I is a collection of random variables {X1 : 2 e I} on a probability space (Cl, ., P) taking values in a set S. The set S is called the state space of the process. In the above, one may take, respectively: (i) I = Z , S = I!; (ii) I = [0, oo), S = Z; (iii) I = l, S = X83 . For the most part we shall study stochastic processes indexed by a one-dimensional set of real numbers (e.g., time). Here the natural ordering of numbers coincides with the sense of evolution of the process. This order is lost for stochastic processes indexed by a multidimensional parameter; such processes are usually referred to as random fields. The state space S will often be a set of real numbers, finite, countable, (i.e., discrete) or uncountable. However, we also allow for the possibility of vector-valued variables. As a matter of convenience in notation the index set is often suppressed when the context makes it clear. In particular, we often write {X„} in place of {X„: n = 0, 1, 2, ...} and {X,} in place of {X,: t >, 0}. For a stochastic process the values of the random variables corresponding
- 21. 2 RANDOM WALK AND BROWNIAN MOTION to the occurrence of a sample point co e fl constitute a sample realization of the process. For example, a sample realization of the coin-tossing process corresponding to the occurrence of w e f2 is of the form (X0 (aw), X,(co), ... , X„(w), ...). In this case X(w) = 1 or 0 depending on whether the outcome of the nth toss is a head or a tail. In the general case of a discrete-time stochastic process with state-space S and index set I = 7L + = {0, 1, 2, ...}, the sample realizations of the process are of the form (X0 (a ), Xl (co), ... , X„(w), ...), X(co) e S. In the case of a continuous-parameter stochastic process with state space S and index set I = I{B + = [0, cc), the sample realizations are functions t —► X(w) e S, w e S2. Sample realizations of a stochastic process are also referred to as sample paths (see Figures 1.1 a, b). In the so-called canonical choice for f) the sample points of f represent sample paths. In this way S2 is some set of functions w defined on I taking values in S, and the value X,(co) of the process at time t corresponding to the outcome co E S2 is simply the coordinate projection X,(w) = coy. Canonical representations of sample points as sample paths will be used often in the text. Stochastic models are often specified by prescribing the probabilities of events that depend only on the values of the process at finitely many time points. Such events are called finite-dimensional events. In such instances the probability measure P is only specified on a subclass ' of the events contained in a sigmafield F. Probabilities of more complex events, for example events that depend on the process at infinitely many time points (infinite-dimensional events), are (a) S (b) Figure 1.1
- 22. THE SIMPLE RANDOM WALK 3 frequently calculated in terms of the probabilities of finite-dimensional events by passage to a limit. The ideas contained in this section will be illustrated in the example and in exercises. Example 1. The sample space S2 for repeated (and unending) tosses of a coin may be represented by the sequence space consisting of sequences of the form w = (col , w2 ,. . . , wn , ...) with awn = 1 or co,, = 0. For this choice of 0, the value of X. corresponding to the occurrence of the sample point w e f is simply the nth coordinate projection of w; i.e., X(w) = w,. Suppose that the probability of the occurrence of a head in a single toss is p. Since for any number n of tosses the results of the first n — 1 tosses have no effect on the odds of the nth toss, the random variables X1 ,. . . , X. are, for each n >, 1, independent. Moreover, each variable has the same (Bernoulli) distribution. These facts are summarized by saying that {X1 , X2,. . .} is a sequence of independent and identically distributed (i.i.d.) random variables with a common Bernoulli distribution. Let Fn denote the event that the specific outcomes E 1 , ... , en occur on the first n tosses respectively. Then Fn = {X1 =e ,...,Xn =En} = {w a fl: w1 = s13 ...,CJn =En } is a finite-dimensional event. By independence, P(F,,) = p'"( 1 — p)" -'" (1.1) where rn is the number of l's among e, . . . , en . Now consider the singleton event G corresponding to the occurrence of a specific sequence of outcomes c ,e ,...,sn ,... . Then G = {Xl =e ,. . . , Xn = En,. . .} = {(E1, E2, . . . , en , ...)} consists of the single outcome a = (E,, e2 , ... , En, ...) in f2. G is an infinite-dimensional event whose probability is easily determined as follows. Since G c Fn for each n > 1, it follows that 0 < P(G) < P(F,,) = p'"(1 — p)" - '" for each n = 1, 2, .... (1.2) Now apply a limiting argument to see that, for 0 < p < 1, P(G) = 0. Hence the probability of every singleton event in S2 is zero. 2 THE SIMPLE RANDOM WALK Think of a particle moving randomly among the integers according to the following rules. At time n = 0 the particle is at the origin. At time n = 1 it
- 23. 4 RANDOM WALK AND BROWNIAN MOTION moves either one unit forward to + I or one unit backward to —1, with respective probabilities p and q = 1 — p. In the case p = 2, this may be accomplished by tossing a balanced coin and making the particle move forward or backward corresponding to the occurrence of a "head" or a "tail", respectively. Similar experiments can be devised for any fractional value of p. We may think of the experiment, in any case, as that of repeatedly tossing a coin that falls "head" with probability p and shows "tail" with probability I — p. At time n the particle moves from its present position S„_ 1 by a unit distance forward or backward depending on the outcome of the nth toss. Suppose that X. denotes the displacement of the particle at the nth step from its position S„_, at time n — 1. According to these considerations the displacement (or increment) process {X.} associated with {S„} is an i.i.d. sequence with P(X„ = + 1) = p, P(X„ _ —1) = q = 1 — p for each n > 1. The position process {S„} is then given by S,,:=Xi +...+X., S0 =0. (2.1) Definition 2.1. The stochastic process {S,,: n = 0, 1, 2, ...} is called the simple random walk. The related process S = S„ + x, n = 0, 1, 2, ... is called the simple random walk starting at x. The simple random walk is often used by physicists as an approximate model of the fluctuations in the position of a relatively large solute molecule immersed in a pure fluid. According to Einstein's diffusion theory, the solute molecule gets kicked around by the smaller molecules of the fluid whenever it gets within the range of molecular interaction with fluid molecules. Displacements in any one direction (say, the vertical direction) due to successive collisions are small and taken to be independent. We shall return to this physical model in Section 7. One may also think of X,, as a gambler's gain in the nth game of a series of independent and stochastically identical games: a negative gain means a loss. Then Sö = x is the gambler's initial capital, and S„ is the capital, positive or negative, at time n. The first problem is to calculate the distribution of S. To calculate the probability of {S; = y}, count the number u of + I's in a path from x to y in n steps. Since n — u is then the number of — l's, one must have u — (n — u) = y — x, or u = (n + y — x)/2. For this, nand y — x must be both even or both odd, and ly — xj <, n. Hence n n + y — x pin+Y—x)12q(n—Y+x)/2 if ly — xI < ri P(S. =Y)= 2 and y — x, n have the same parity, 0 otherwise. (2.2)
- 24. TRANSIENCE AND RECURRENCE PROPERTIES OF THE SIMPLE RANDOM WALK 5 3 TRANSIENCE AND RECURRENCE PROPERTIES OF THE SIMPLE RANDOM WALK Let us first consider the manner in which a particle escapes from an interval. Let TY denote the first time that the process starting at x reaches y, i.e. Ty:= min{n >, 0:S„ = y}. (3.1) To avoid trivialities, assume 0 <p < 1. For integers c and d with c < d, denote 4(x):= P(T < T' ). (3.2) In other words, 4(x) is the probability that the particle starting at x reaches d before it reaches c. Since in one step the particle moves to x + I with probability p, or to x — 1 with probability q, one has 4(x) = po(x + 1) + q4(x — 1) (3.3) so that O(x+ 1)—O(x)=-[O(x)—cß(x— 1)], c+ 1 ,<x,<d— 1 p 0(c) = 0, (3.4) q(d) = 1. Thus, ¢(x) is the solution to the discrete boundary-value problem (3.4). For p ^ q, Eq. 3.4 yields x-1 x-1 q Y O(x) = Z [^(y + 1 ) — o(y)] = Z - [O(c + 1) — O(c)] v=c v=c P x -' =0(c+1) Y - =0(c+1) 1 1- - (q/P) /p)(3.5) yc ' P) To determine 4(c + 1) take x = d in Eq. 3.5 to get 1 =4(d)=4(c+ 1) 1 — (qlP)°-` 1 — q/P Then 1 — q/P q(c + 1) = 1 — (glp)d c
- 25. 6 RANDOM WALK AND BROWNIAN MOTION so that P(Tx<Tx)= 1 —(q/P)x for c<x<d, p q. (3.6) 1 — (q/P)d-c Now let 0/i(x)==P(T, < Tdx). (3.7) By symmetry (or the same method as above), P(Tx<Td)= 1—(P/q)d-xfor c<x<d,p q. (3.8) 1 — (P/q) d-c Note that O(x) + fr(x) = 1, proving that the particle starting in the interior of [c, d] will eventually reach the boundary (i.e., either c or d) with probability 1. Now if c < x, then (Exercise 3) P({S„} will ever reach c) = P(T,' < oo) = lim i(x) d-+oo x — • um 9/ ifp 1 >2 = dam°°— P c 1, ifp <Z, x—c q if p> = P(3.9) 1, ifp<Z. By symmetry, or as above, i P({S:} will ever reach d) = P(Td' < oo) = 1' d_x f p > 2 (3.10) C i q/ , f p < Z . Observe that one gets from these calculations the (geometric) distribution function for the extremes Mx = sup,, S„ and mx = inf„ S; (Exercise 7). Note that, by the strong law of large numbers (Chapter 0), x P S„ = x+S„ ^p—gasn--•oo =1. (3.11) n n
- 26. TRANSIENCE AND RECURRENCE PROPERTIES OF THE SIMPLE RANDOM WALK 7 Hence, if p > q, then the random walk drifts to + oo (i.e., S„ -* + co) with probability 1. In particular, the process is certain to reach d > x if p > q. Similarly, if p < q, then the random walk drifts to - co (i.e., S„ -+ - cc), and starting at x > c the process is certain to reach c if p < q. In either case, no matter what the integer y is, P(Sn = y i.o.) = 0, if p q, (3.12) where i.o. is shorthand for "infinitely often." For if Sx = y for integers nl < n2 < • • through a sequence going to infinity, then x = y -+ 0 as nk -• cc, nk nk the probability of which is zero by Eq. 3.11. Definition 3.1. A state y for which Eq. 3.12 holds is called transient. If all states are transient then the stochastic process is said to be a transient process. In the case p = q = 2, according to the boundary-value problem (3.4), the graph of 4(x) is along the line of constant slope between the points (c, 0) and (d, 1). Thus, Similarly, Again we have P(Tx<Tx)= x-c , d-c P(Tcx <Td)= d-x d-c c<x<d,p=q =Z c<,x<d,p=q =2 (3.13) (3.14) q5(x) + i(x) = 1. (3.15) Moreover, in this case, given any initial position x> c, P({S„} will eventually reach c) = P(Tc < cc) = lim P({S„} will reach c before it reaches d) = lim d-x = 1. (3.16) d - mo d - c
- 27. 8 RANDOM WALK AND BROWNIAN MOTION Similarly, whatever the initial position x < d, P({S.} will eventually reach d) = P(Td < oo) = lim x—c = 1. (3.17) e -- ao d — c Thus, no matter where the particle may be initially, it will eventually reach any given state y with probability 1. After having reached y for the first time, it will move to y + 1 or to y — 1. From either of these positions the particle is again bound to reach y with probability 1, and so on. In other words (Exercise 4), P(S' = y i.o.) = 1, if p = 9 = 2. (3.18) This argument is discussed again in Example 4.1 of Chapter II. Definition 3.2. A state y for which Eq. 3.18 holds is called recurrent. If all states are recurrent, then the stochastic process is called a recurrent process. Let r^X denote the time of the first return to x, rl„ := inf{n >, 1: S.' = x} . (3.19) Then, conditioning on the first step, it will follow (Exercise 6) that P(f],, < oo) = 2 min(p, q). (3.20) 4 FIRST PASSAGE TIMES FOR THE SIMPLE RANDOM WALK Consider the random variable 7 := T° representing the first time the simple random walk starting at zero reaches the level (state) y. We will calculate the distribution of T y by means of an analysis of the sample paths of the simple random walk. Let FN,y = {Ty = N} denote the event that the particle reaches state y for the first time at the Nth step. Then, FN.y ={S„iky for n=0,1,...,N-1,SN =y}. (4.1) Note that "SN = y" means that there are (N + y)/2 plus l's and (N — y)/2 minus 1's among XI , X2 , ... , XN (see Eq. 2.1). Therefore, we assume that IYI <, N and N + y is even. Now there are as many paths leading from (0, 0) to (N, y) as there are ways of choosing (N + y)/2 plus l's among X1 , X2 , ... , XN , namely N N+y 2
- 28. FIRST PASSAGE TIMES FOR THE SIMPLE RANDOM WALK 9 Each of these choices has the same probability of occurrence, specifically p(N+y)/2q(N-r)/z• Thus, P(FN.Y) = Lp(N+r)rzq(N -v»z (4.2) where L is the number of paths from (0, 0) to (N, y) that do not touch or cross the level y prior to time N. To calculate L, consider the complementary number L of paths that do reach y prior to time N, N L'= N+y —L. (4.3) 2 First consider the case of y> 0. If a path from (0, 0) to (N, y) has reached y prior to time N, then either (a) SJY _ 1 = y + 1 (see Figure 4.1a) or (b) SN _ 1 = y — I and the path from (0, 0) to (N — 1, y — 1) has reached y prior to time N — 1 (see Figure 4.1b). The contribution to L from (a) is N-1 N+y 2 We need to calculate the contribution to L from (b). y (a) 1I Figure 4.1
- 29. 10 RANDOM WALK AND BROWNIAN MOTION Proposition 4.1. (A Reflection Principle). Let y > 0. The collection of all paths from (0, 0) to (N — 1, y — 1) that touch or cross the level y prior to time N — 1 is in one-to-one correspondence with the collection of all possible paths from (0,0)to (N — 1,y + 1). Proof. Given a path y from (0, 0) to (N — 1, y + 1), there is a first time r at which the path reaches level y. Let y' denote the path which agrees with y up to time T but is thereafter the mirror reflection of y about the level y (see Figure 4.2). Then y' is a path from (0, 0) to (N — 1, y — 1) that touches or crosses the level y prior to time N — 1. Conversely, a path from (0, 0) to (N — 1, y — 1) that touches or crosses the level y prior to time N — 1 may be reflected to get a path from (0, 0) to (N — 1, y + 1). This reflection transformation establishes the one-to-one correspondence. n It now follows from the reflection principle that the contribution to L' from (b) is N-1 N+y 2 Hence N-1 L' =2 N+y (4.4) 2 Therefore, by (4.3), (4.2), N N-1 P(T,,=N)= P(FN.y)= N + y — 2 N + y p(N +Y)/2q(N-y)/2 2 2 N- Figure 4.2
- 30. MULTIDIMENSIONAL RANDOM WALKS 11 N = IYI N + y p(N+Y)12q(N_Y)1i for N >, y, y + N even, y > 0 N 2 (4.5) To calculate P(TT = N) for y < 0, simply relabel H as T and T as H (i.e., interchange + 1, — 1). Using this new code, the desired probability is given by replacing y by —y and interchanging p, q in (4.5), i.e., N P(Tr = N) = —( + y q(N_Y)/2p(N+Y)/2 2 Thus, for all integers y 0, one has N P(Ty = N) = N + y p(N+y)/2q(x -v)I2 = I p(SN = y) (4.6) N 2 for N = IYI, IYI + 2, IYI + 4, .... In particular, if p = q = Z, then (4.6) yields N P(Ty = N)= IN N+y ZNfor N= IYI,IYI +2,IYI +4,.... (4.7) 2 However, observe that the expected time to reach y is infinite since by Stirling's formula, k! = (2irk) 1 / 2Ve - '`( 1 + o(1)) as k -,, oo, the tail of the p.m.f. of Ty is of the order of N -3/2 as N -• oo (Exercise 10). 5 MULTIDIMENSIONAL RANDOM WALKS The k-dimensional unrestricted simple symmetric random walk describes the motion of a particle moving randomly on the integer lattice 7Lk according to the following rules. Starting at a site x = (x., ... , xk ) with integer coordinates, the particle moves to a neighboring site in one of the 2k coordinate directions randomly selected with probability 1/2k, and so on, independently of previous displacements. The displacement at the nth step is a random variable X. whose possible values are vectors of the form ±e ;, i = 1, ... , k, where the jth component of e; is 1 for j = i and 0 otherwise. X 1 , X2 ,... are i.i.d. with P(X„= e;)=P(X„= —e,) = 1/2k fori= 1,...,k. (5.1)
- 31. 12 RANDOM WALK AND BROWNIAN MOTION The corresponding position process is defined by Sö=x, S"= x+X 1 +•••+Xn , ni1. (5.2) The case k = 1 is that already treated in the preceding sections with p = q = 2. In particular, for k = 1 we know that the simple symmetric random walk is recurrent. Consider the coordinates of X. = (X,.. . , X.). Although X,', and X„ are not independent, notice that they are uncorrelated for i 0 j. Likewise, it follows that the coordinates of the position vector S = (Sn' 1 , ... , S) are uncorrelated. In particular, ES„ = x, xi xj if =j Cov(Sn ' , Sn') = tn, i (5.3) 0, ifi*j. Therefore the covariance matrix of S. is nI where I is the k x k identity matrix. The problem of describing the recurrence properties of the simple symmetric random walk in k dimensions is solved by the following theorem of Pö1ya. Theorem 5.1. [P61ya]. {S„} is recurrent for k = 1, 2 and transient for k ? 3. Proof. The result has already been obtained for k = 1. In general, let S. = Sn o and write rn=P(Sn=0) fn = P(S" = 0 for the first time after time 0 at n), n >, 1. (5.4) Then we get the convolution equation n rn = fj r"_jforn=1,2,..., j=0 ro=1, f0= 0 . (5.5) Let P(s) and f(s) denote the respective probability generating functions of {rn} and {f,.} defined by P(s) _ f (s) _ > fn s" (0 < s < 1). (5.6) n =o n =o The convolution equation (5.5) transforms as P(s) = 1 + I E .ijrn-js'sn-j = 1 + Z (MW =0 Y rm sm)f j sj = 1 + f(s)f(s). (5.7) n=1j =0 j =0
- 32. MULTIDIMENSIONAL RANDOM WALKS 13 Therefore, r(s)1 —f(s) (5.8) The probability of eventual return to the origin is given by 00 Y:= Y- fn =.f(l)• (5.9) Note that by the Monotone Convergence Theorem (Chapter 0), P(s) ,, r(1) and f(s) / f(1) as s T 1. If f(1) < 1, then P(l) = (1 — f(1))' < oo. If f(1) = 1, then P(1) = ums , (1 — f(s) = oo. Therefore, y < 1 (i.e., 0 is transient) if and only if ß:=r(1) < oo. This criterion is applied to the case k = 2 as follows. Since a return to 0 is possible at time 2n if and only if the numbers of steps among the 2n in the positive horizontal and vertical directions equal the respective numbers of steps in the negative directions, " (2n)! 1 2n ( n2 r2n=4 -in =_ j=o j!j!(n — j)!(n — j)! 42" n j 41 2n n n^ n^I n n j4 1n n^z . (5.10) 2n j=o The combinatorial identity used to get the last line of (5.10) follows by considering the number of ways of selecting samples of size n from a population of n objects of type 1 and n objects of type 2 (Exercise 2). Apply Stirling's formula to (5.10) to get r2 = 0(1/n) > c/n for some c > 0. Therefore, ß = P(1) = + oo and so 0 is recurrent in the case k = 2. In the case k = 3, similar considerations of "coordinate balance" give rzn = 6—zn (2n)! (j.m):j+m ,n j!j!m!m!(n — j — m)!(n — j — m)! 1 ( 2n) 1 n! 2 = 22n n j+msn 3" j!m!(n — j — m)!} . (5.11) Therefore, writing n!1 pj,m =— j!m!(n — j — m)! 3n and noting that these are the probabilities for the trinomial distribution, we have
- 33. 14 RANDOM WALK AND BROWNIAN MOTION that 1 (2n) = z" (P;.m) 2 (5.12) 2 n is nearly an average of pj,m's (with respect to the pj,m distribution). In any case, 22n (2n) jmax Pj,m]Pj,m= 2a" ( n n) ma x Pj.m. (5.13) j,m j,m The maximum value of pj,m is attained at j and m nearest to n/3 (Exercise 5). Therefore, writing [x] for the integer part of x, i r2n 1( 2n) 1 n. (5.14) 22n n 3" rn i fl, [n], Apply Stirling's formula to get (see 5.19 below), r 2n - C 2" nn n n 3/2 for some C' > 0. (5.15) In particular, Er"<oo. (5.16) " The general case, r2n < ck n-k/2 for k > 3, is left as an exercise (Exercise 1). n The constants appearing in the estimate (5.15) are easily computed from the monotonicity of the ratio n!/{(2nn)`I2n"e- "}; whose limit as n -> oo is 1 according to Stirling's formula. To see that the ratio is monotonically decreasing, simply observe that t logn!= log n! — flog n — n log n + n — log(2n) li 2 (2nn)112n"e- " J.j log j— Z log n}—{n logn—n}—log(2n)"2 ,-1 ) = j log(j — 1) + log(j) — f " log x dx } + 1 — log(2n)l iz ( U2 2 J^ J (5.17)
- 34. CANONICAL CONSTRUCTION OF STOCHASTIC PROCESSES 15 where the integral term may be checked by integration by parts. The point is that the term defined by " log(J — 1) + log(j) (5.18) j =2 2 provides the inner trapezoidal approximation to the area under the curve y = log x, 1 < x <, n. Thus, in particular, a simple sketch shows 01 J logxdx—T" is monotonically increasing. So, in addition to the asymptotic value of the ratio, one also has n! e 1(2nn)112 n"e -" < (2n)1 / z , n = 1, 2, .... (5.19) 6 CANONICAL CONSTRUCTION OF STOCHASTIC PROCESSES Often a stochastic process is defined on a given probability space as a sequence of functions of other already constructed random variables. For example, the simple random walk {S" = Xl + • • • + X"}, So = 0 is defined in terms of the coin-tossing process {X"} in Section 2. At other times, a probability space is constructed specifically to define the stochastic process. For example, the probability space for the coin-tossing process was constructed starting from the specifications of the probabilities of finite sequences of heads and tails. This latter method, called the canonical construction, is elaborated upon in this section. Consider the case that the state space is R' (or a subset of it) and the parameter is discrete (n = 1, 2, ...). Take S2 to be the space of all sample paths; i.e., 52:= (ff!)' := R' is the space of all sequences co = (cw l , w2 ,...) of real numbers. The appropriate sigmafield .y := R°° is then the smallest sigmafield containing all finite-dimensional sets of the form {w e SZ: w 1 e B I , ... , w e Bk }, where BI , ... , Bk are Borel subsets of W. The coordinate functions X" are defined by X(w) = con. As in the case of coin tossing, the underlying physical process sometimes suggests a specification of probabilities of finite-dimensional events defined by the values of the process at time points 1, 2, ... , n for each n >, 1. That is, for each n > 1 a probability measure P. is prescribed on (R", M"). The problem is that we require a probability measure P on (f, F) such that P" is the distribution of X1 , ... , X. That is, for all Borel sets B 1 , ... , B", P(we02:m,EB 1 ,...,Cw"EB")=P"(B 1 x ... x B"). (6.1)
- 35. 16 RANDOM WALK AND BROWNIAN MOTION Equivalently, P(X1 E B 1 , ... , X„ E B„) = P„(B 1 x • .. x B„). (6.2) Since the events {X1 c-B1 ,...,X„EB„,Xn+1 eR'}and {X1 eB1 ,...,X„EB„} are identical subsets of .^'°°, for there to be a well-defined probability measure P prescribed by (6.1) or (6.2) it is necessary that PP +1(B, x • • • x B„ x Ili') = Pn (B, x • .. x B„)(6.3) for all Borel sets B 1 , . .. , B. in 118 1 and n >, 1. Kolmogorov's Existence Theorem asserts that the consistency condition (6.3) is also sufficient for such a probability measure P to exist and that there is only one such P on (, R) = (12, F) (theoretical complement 1). This holds more generally, for example, when the state space S is l, a countable set, or any Borel subset of tF . A proof for the simple case of finite state processes is outlined in Exercise 3. Example 1. Consider the problem of canonically constructing a sequence X1 , X2 , ... of i.i.d. random variables having the common (marginal) distribution Q on (1111 , R1 ). Take il = IR', F = R', and X. the nth coordinate projection X(w) = w„, w E S2. Define, for each n >, 1 and all Borel sets B1 , ... , B,,, p,, (B1 x ... x B.) = Q(B 1 ). . . Q(B,,). (6.4) Since Q(R') = 1, the consistency condition (6.3) follows immediately from the definition (6.4). Now one simply invokes the Kolmogorov Existence Theorem to get a probability measure P on (S2, F) such that P(X1 E B1 , ... , Xq E Bn) = Q(B1) ... Q(Bn) = p(X1 EB1) . . .p(X,,EB„). (6.5) The simple random walk can be constructed within the framework of the canonical probability space (S2, F, P) constructed for coin tossing, although this is a noncanonical probability space for {S„}. Alternatively, a canonical construction can be made directly for {S„} (Exercise 2(i)). This, on the other hand, provides a noncanonical probability space for the displacement (coin-tossing) process defined by the differences X. = S. — S„_ 1 , n > 1. Example 2. The problem is to construct a Gaussian stochastic process having prescribed means and covariances. Suppose that we are given a sequence of real numbers µi , pa ,... , and an array, a1 , i, j = 1, 2, ... , of real numbers satisfying (Symmetry) Qi; = o;ifor all i, j, (6.6)
- 36. BROWNIAN MOTION 17 (Non-negative Definiteness) Z 6;j x; xj ^ 0 for all n-tuples (x 1 , ... , x„) in (6.7) i,j= 1 Property (6.7) is the condition that D. = ((Q ;j )), ,; , be a nonnegative definite matrix for each n. Again take ) = R', = :4', and X,, X2 ,. . . the respective coordinate projections. For each n >, 1, let P„ be the n-dimensional Gaussian distribution on (O", ") having mean vector (µ l , . .. , µ„) and covariance matrix D. Since a linear transformation of a Gaussian random vector is also Gaussian, the consistency condition (6.3) can be checked by applying the coordinate projection mapping (x 1 ..... x„ + ,) -+ (x 1 , ... , x„) from I ”+' to tll” (Exercise 1). Example 3. Let S be a countable set and let p = ((p)) be a matrix of nonnegative real numbers such that for each fixed i, p;,j is a probability distribution (sums to I over j in S). Let a = (7r i ) be a probability distribution on S. By the Kolmogorov Existence Theorem there is a probability distribution P,, on the infinite sequence space = S x S x • • • x S x • • . such that PP (X0 =10.....X =j) = n30 pJ0 J1 • • •p where X„ denotes the nth projection map (Exercise 2(ii)). In this case the process {X„}, having distribution P, is called a Markov chain. These processes are the subject of Chapter II. 7 BROWNIAN MOTION Perhaps the simplest way to introduce the continuous-parameter stochastic process known as Brownian motion is to view it as the limiting form of an unrestricted random walk. To physically motivate the discussion, suppose a solute particle immersed in a liquid suffers, on the average, f collisions per second with the molecules of the surrounding liquid. Assume that a collision causes a small random displacement of the solute particle that is independent of its present position. Such an assumption can be justified in the case that the solute particle is much heavier than a molecule of the surrounding liquid. For simplicity, consider displacements in one particular direction, say the vertical direction, and assume that each displacement is either +A or —A with probabilities p and q = 1 — p, respectively. The particle then performs a one-dimensional random walk with step size A. Assume for the present that the vessel is very large so that the random walk initiated far away from the boundary may be considered to be unrestricted. Suppose at time zero the particle is at the position x relative to some origin. At time t > 0 it has suffered approximately n = tf independent displacements, say Z 1 , Z2 , ... , Z„. Since f is extremely large (of the order of 10 21 ), if t is of the order of 10 -10 second then n is very large. The position of the particle at time t, being x plus the sum of n independent Bernoulli random variables, is, by the central limit theorem, approximately Gaussian with mean x + tf(p — q)0 and variance tf4A2pq. To make the limiting argument firm, let
- 37. 18 RANDOM WALK AND BROWNIAN MOTION p=2+ 2^ o and 0= ^ Here p and a are two fixed numbers, or > 0. Then as f --> cc, the mean displacement t f (p — q)0 converges to tµ and the variance converges to tae . In the limit, then, the position X, of the particle at time t > 0 is Gaussian with probability density function (in y) given by _ z P(t; x, Y) _ (2ita2t)1j2 eXp{— (Y _ 2QZt tµ) (7.1) Ifs > 0 then X, + — X, is the sum of displacements during the time interval (t, t + s]. Therefore, by the argument above, X, +s — X, is Gaussian with mean sµ and variance sae, and it is independent of {X,,: 0 < u < t}. In particular, for every finite set of time points 0 < tl < t2 < • • • <t, the random variables X,,, X^2 — X,.. . , X XX,,,_, are independent. A stochastic process with this last property is said to be a process with independent increments. This is the continuous-time analogue of random walks. From the physical description of the process {X} as representing (a coordinate of) the path of a diffusing solute particle, one would expect that the sample paths of the process (i.e., the trajectories t —* X(w) = w,) may be taken to be continuous. That this is indeed the case is an important mathematical result originally due to Norbert Wiener. For this reason, Brownian motion is also called the Wiener process. A complete definition of Brownian motion goes as follows. Definition 7.1. A Brownian motion with drift µ and diffusion coefficient a2 is a stochastic process {X,: t 0} having continuous sample paths and independent Gaussian increments with mean and variance of an increment XX+s — XX being sp and sae , respectively. If X0 = x, then this Brownian motion is said to start at x. A Brownian motion with zero drift and diffusion coefficient of 1 is called the standard Brownian motion. Families of random variables {X} constituting Brownian motions arise in many different contexts on diverse probability spaces. The canonical model for Brownian motion is given as follows. 1. The sample space S2 := C[0, oo) is the set of all real-valued continuous functions on the time interval [0, cc). This is the set of all possible trajectories (sample paths) of the process. 2. XX (co) := co, is the value of the sample path w at time t. 3. S2 is equipped with the smallest sigmafield .y of subsets of S2 containing the class .moo of all finite-dimensional sets of the form F = fce e ): a; <w,. < bi , i = 1, 2, ... , k}, where a; <b, are constants and 0 < t l < t2 < • • • < tk are a finite set of time points. .F is said to be generated by .moo.
- 38. BROWNIAN MOTION 19 4. The existence and uniqueness of a probability measure Px on F, called the Wiener measure starting at x, as specified by Definition 7.1 is determined by the probability assignments of the form of (7.2) below. For the set F above, PP (F) can be calculated as follows. Definition (7.1) gives the joint density of X, Xr2 - X,,, ... , X,, - Xtk _, as that of k independent Gaussian random variables with means t 1 p., (t2 - t1)µ, ... , (tk - tk -1)/2 , respectively, and variances tIQ 2 , (t2 — t1)a 2 , ... , (tk — tk_ 1 )a2 , respectively. Transforming this (product) joint density, say in variables z 1 , z2 , ... , by the change of variables z 1 = YI, z2 = Y2 - Y1' • • • I zk = Yk - Yk-1 and using the fact that the Jacobian of this linear transformation is unity, one obtains PX (a; <X11 <b1 fori= 1,2,...,k) ^ bI ... fbk - I fbk {—(Y1 — X — tA' 2 01ak I Jak (27rQ 2 t l) "2 exp 2v t, 1Y2 — Y1 — (2 — t1)11)2 l 1 t (21IU2(t2 — t1))"2exp — 2a2(t2 — t1) 1 I J (Yk — Yk-1 — (tk — tk-1)t^)2 — tk - 1)) 1/ ^ ... 2 2 expl(— 2 2 (tk — tk - 1) dYk dYk-I ... dY1• (27L6 (tk 6 (7.2) The joint density of X, 1 , X,2 , ... , X,,, is the integrand in (7.2) and may be expressed, using (7.1), as P(t1;x+Y1)P(t2 — t1+Y1 , Y2)"'P(tk — tk-I+Yk-I,Yk)• (7.3) The probabilities of a number of infinite-dimensional events will be calculated in Sections 9-13, and in Chapter IV. Some further discussion of mathematical issues in this connection are presented in Section 8 also. The details of a construction of the Brownian motion and its Wiener measure distribution are given in the theoretical complements of Section 13. If {X,(') }, j = 1, 2, ... , k, are k independent standard Brownian motions, then the vector-valued process {X,} = {{X; 1) , Xt(2I, ... , Xr(k) )} is called a standard k-dimensional Brownian motion. If {X,} is a standard k-dimensional Brownian motion, p = (µ(1) , ... , µ(k1 ) a vector in l, and A a k x k nonsingular matrix, then the vector-valued process {Y, = AX, + tµ} has independent increments, the increment Y +S - Y, = A(X, +s - X,) + (t + s - t)µ being Gaussian with mean vector su and covariance (or dispersion) matrix sD, where D = AA' and A' denotes the transpose of A. Such a process Y is called a k-dimensional Brownian motion with drift vector p and diffusion matrix, or dispersion matrix D.
- 39. 20 RANDOM WALK AND BROWNIAN MOTION 8 THE FUNCTIONAL CENTRAL LIMIT THEOREM (FCLT) The argument in Section 7 indicating Brownian motion (with zero drift parameter for simplicity) as the limit of a random walk can be made on the basis of the classical central limit theorem which applies to every i.i.d. sequence of increments {Zm} having finite mean and variance. While we can only obtain convergence of the finite-dimensional distributions by such considerations, much more is true. Namely, probabilities of certain infinite-dimensional events will also converge. The convergence of the full distributions of random walks to the full distribution of the Brownian motion process is informally explained in this section. A more detailed and precise discussion is given in the theoretical complements of Sections 8 and 13. To state this limit theorem somewhat more precisely, consider a sequence of i.i.d. random variables {Z.} and assume for the present that EZ,„ = 0 and Var Z. = a 2 > 0. Define the random walk S0= 0 ,Sm =Z1 +•.•+.Zm(m=1,2,...). (8.1) Define, for each value of the scale parameter n >, 1, the stochastic process Xin) = S[n ^ r](t i 0), (8.2) V " where [nt] is the integer part of nt. Figure 8.1 plots the sample path of {X;"^: t >, 0} up to time t = 13/n if the successive displacements take values Z1 =-1, Zz =+1, Z3 =+1, Z4 =+1, Z5 =-1, Z6 =+1, Z7 =+1, Zg = — 1, Zq = + 1, Z10 = + 1, 211 = + 1, Z12 = — 1. _ 1 Simi .—i ‚In 4 —4 '-4 Vn 3 ^---; •-4 'In ? —4—.--1-4 Vn Vn 1 1 3 4 5 6 7 i 9 10 11 12 13 t n n n n n n It n n n n n n Intl EX' ") = 0, VarX^11 = n ~ t, Cov(Vt., + `v l)= [ n] ~ S. Figure 8.1
- 40. THE FUNCTIONAL CENTRAL LIMIT THEOREM (FCLT) 21 The process {S11 : t >, 0} records the discrete-time random walk {S.: m = 0, 1, 2, ...} on a continuous time scale whose unit is n times that of the discrete time unit, i.e., Sm is plotted at time m/n. The process {X} = {(1//)S[„f] } also scales distance by measuring distances on a scale whose unit is f times the unit of measurement used for the random walk. This is a convenient normalization, since EX ) = 0, Var X}") = z [nt]0 2 tazfor large n. (8.3) n In a time interval (t 1 , t2] the overall "displacement" X — X(°) is the sum of a large number [ntz] — [nt,] n(t2 — t,) of small i.i.d. random variables 1 1 In the case {Z,„} is i.i.d. Bernoulli, this means reducing the step sizes of the random variables to t1 = 1/,.fn. In a physical application, looking at {X} means means the following. 1. The random walk is observed at times t, < t2 <t3 < • • • sufficiently far apart to allow a large number of individual displacements to occur during each of the time intervals (t,, tz], (tz, t3], ... , and 2. Measurements of distance are made on a "macroscopic" scale whose unit of measurement is much larger than the average magnitude of the individual displacements. The normalizing large parameter n scales time and n'' z scales space coordinates. Since the sample paths of {X} have jumps (though small for large n) and are, therefore, discontinuous, it is technically more convenient to linearly interpolate the random walk between one jump point and the next, using the same space—time scales as used for {X°}. The polygonal process {X,(" 1 } is formally defined by Xt") = SIntl + (nt — [nt])t 0. (8.4) In this way, just as for the limiting Brownian motion process, the paths of {X} are continuous. Figure 8.2 plots the path of {X1"°} corresponding to the path of {X} drawn in Figure 8.1. In a time interval m/n < t < (m + 1)/n, X;") is constant at level 1// S„„ while X}") changes linearly from l/ f S. at time t=m/n to I S„, +1 = S "' Z'" + ' at time m + 1 me t = n
- 41. 22 RANDOM WALK AND BROWNIAN MOTION I [n(] Z101j+j S^ rl + (t — n ) ^n Vn 4 Vn 3 ‚In 2 do W, = 0, VarX^rn) _ [nt] + 1 (t — [nt] )2 t n n n [ns] Figure 8.2 Thus, in any given interval [0, T] the maximum difference between the two processes {X,(n»} and {X,(°) } does not exceed c n (T) = max IZII , IZ21 , IZ[ nT,+1I To see that the difference between {X,(n) } and {X;n) } is negligible for large n, consider the following estimate. For each 6 > 0, P(en (T) > 6) = 1 — P(en (T) < (5) = I —P( IZ^'<(5 for allm= 1,2,...,[nT]+ 1) l V n = 1 — (P(IZ11 < 5)) [nT1+1 = 1 — (1 — P(IZ11 > 6.^ n))[nT1 +l (8.5) Assuming for simplicity that EIZ1 13 < co, Chebyshev's inequality yields P(1Z11 > 6^) <, EIZII 3/63n3/2 . Use this in (8.5) to get (Exercise 9) EIZIr (nTJ+l P(e (T) > (5) 1 — ( i — (533/2 ) 1—exp{— EIZ113T }—► 0 (8.6) 63n1/2 when n is large. Here indicates that the difference between the two sides
- 42. THE FUNCTIONAL CENTRAL LIMIT THEOREM (FCLT) 23 goes to zero. Thus, on any closed and bounded time interval the behaviors of {X,(") } and {X} are the same in the large-n limit. Note that given any finite set of time points 0 < t 1 < t2 < < t, the joint distribution of (X, X;z) , .. . , X(") ) converges to the finite-dimensional distribution of (XX1 , X,2 , ... , Xtk ), where {X,} is a Brownian motion with zero drift and diffusion coefficient a 2 . To see this, note that X, X — Xt"^, ... , X,(k ) — X() , are independent random variables that by the classical central limit theorem (Chapter 0) converge in distribution to Gaussian random variables with zero means and variances t1a2 , (t2 — t, )a 2 , ... , (tk — tk_ t )Q 2 . That is to say, the joint distribution of (X, X,(") — X°, X (") — X ("^ )converges to that of (X,,, X X, ... , X X,k _,). By a linear transformation, one gets the desired convergence of finite-dimensional distributions of {X(" ) } (and, therefore, of {X^"1 }) to those of the Brownian motion process {X} (Exercise 1). Roughly speaking, to establish the full convergence in distribution of {X!" 1} to Brownian motion, one further looks at a finite set of time points comprising a fine subdivision of a bounded interval [0, T] and shows that the fluctuations of the process {X^"^} on [0, T] between successive points of this subdivision are sufficiently small in probability, a property called the tightness of the process. This control over fluctuations together with the convergence of {X^"1} evaluated at the time points of the subdivision ensures convergence in distribution to a continuous process whose finite-dimensional distributions are the same as those of Brownian motion (see theoretical complements for details). Since there is no process other than Brownian motion with continuous sample paths that has these limiting finite-dimensional distributions, it follows that the limit must be Brownian motion. A precise statement of the functional central limit theorem (FCLT) is the following. Theorem 8.1. (The Functional Central Limit Theorem). Suppose {Z,,,: m = 1, 2, ...} is an i.i.d. sequence with EZ„, = 0 and variance a 2 > 0. Then as n —* cc the stochastic processes {X: t > 0} (or {Xr"°: t >, 0}) converge in distribution to a Brownian motion starting at the origin with zero drift and diffusion coefficient a 2 . An important way in which to view the convergence asserted in the FCLT is as follows. First, the sample paths of the polygonal process {X^" 1 } belong to the Space S = C[O, oo) of all continuous real-valued function on [0, oo), as do those of the limiting Brownian motion {X}. This space C[O, oo) is a metric space with a natural notion of convergence of sequences {d")}, say, being that "{m(")} converges to w in C[O, co) as n tends to infinity if and only if {co(") (t): a <, t < b} converges uniformly to {w(t): a < t < b} for all closed and bounded intervals [a, b]." Second, the distributions of the processes {X} and {X,} are probability measures P" and P on a certain class F of events of C[0, cc), called Borel subsets, which is generated by and therefore includes all of the finite-dimensional events. .F includes as well various important infinite-
- 43. 24 RANDOM WALK AND BROWNIAN MOTION dimensional events, e.g., the events {max a , b X > y} and fmaxa t b X < x} pertaining to extremes of the process. More generally, if f is a continuous function on C[0, oo) then the event { f({X}) < x} is also a Borel subset of C[0, oo) (Exercise 2). With events of this type in mind, a precise meaning of convergence in distribution (or weak convergence) of the probability measures P. to P on this infinite-dimensional space C[0, oo) is that the probability distributions of the real-valued (one dimensional) random variables f({X;'°}) converge (in distribution as described in Chapter 0) to the distribution of f({X1 }) for each real-valued continuous function f defined on C[0, cc). Since a number of important infinite-dimensional events can be expressed in terms of continuous functionals of the processes, this makes calculations of probabilities possible by taking limits; for examples of infinite dimensional events whose probabilities do not converge see Exercise 9.3(iv). Because the limiting process, namely Brownian motion, is the same for all increments {Z,„} as above, the limit Theorem 8.1 is also referred to as the Invariance Principle, i.e., invariance with respect to the distribution of the increment process. There are two distinct types of applications of Theorem 8.1. In the first type it is used to calculate probabilities of infinite-dimensional events associated with Brownian motion by studying simple random walks. In the second type it (invariance) is used to calculate asymptotics of a large variety of partial-sum processes by studying simple random walks and Brownian motion. Several such examples are considered in the next two sections. 9 RECURRENCE PROBABILITIES FOR BROWNIAN MOTION The first problem is to calculate, for a Brownian motion {X} with drift Ic = 0 and diffusion coefficient Q2 , starting at x, the probability P(T < Ta) = P({X' } reaches c before d) (c < x < d), (9.1) where T;:= inf{t >, 0: Xx = y} . (9.2) Since {B, = (X; — x)/v) is a standard Brownian motion starting at zero, P(2x < ra) = P({B,} reaches c — x before d —_X (9.3) a Q Now consider the i.i.d. Bernoulli sequence {Zm : m = 1, 2, ...} with P(Zm = 1) = P(Zm = — 1) = 2, and the associated random walk So = 0, Sm = Zl + • • • + Z,„ (m >, 1). By the FCLT (Theorem 8.1), the polygonal process {X} associated with this random walk converges in distribution to {B,}. Hence (theoretical
- 44. RECURRENCE PROBABILITIES FOR BROWNIAN MOTION 25 complement 2) c—x d—xl P(i < rd) = lim P( {i} reaches ------- before ----/) "- xQ 6 = lim P({S,„} reaches c" before d"), (9.4) "-+00 where c"= Lc -x ;], 6 and d— x n if d" = d —X is an integer, d = " d—x d x+ 1 if not. By relation (3.14) of Section 3, one has d_ x - - n a P(rx <t) = l d im " = lim ----- -- . (9.5) Therefore, P(r, <ra) = d—c(c<x<d,µ=0). (9.6) Similarly, using relation (3.13) of Section 3 instead of (3.14), one gets P(ta <T') = x_c(c <x<d,p =0). (9.7) Letting d --* + oo in (9.6) and c —+ — co in (9.7), one has P(rc < oo) = P({X, } ever reaches c) = I (c < x, p = 0), (9.8) P(r< < oo) = P({X; } ever reaches d) = 1 (x < d, p = 0). The relations (9.8) mean that a Brownian motion with zero drift is recurrent,
- 45. 26 RANDOM WALK AND BROWNIAN MOTION just as a simple symmetric random walk was shown to be in Section 3. The next problem is to calculate the corresponding probabilities when the drift is a nonzero quantity p. Consider, for each large n, the Bernoulli sequence P(Z,n.n=+1)=Pn=1+ p 2 26^ , {Z,„ n : m = 1, 2, ...} with 1 P(Zm,n= —1)=9n =---------. µ 22 Write Sm,n =Zl ,n +— , +Z,„ n for m>,1,So.,,=0.Then, [nt] µ EX (n) = ES1n,1,n = a n tp ^nor(9.9) Var X^ [ nt] Var Z„n — — [nt] ( (1 1 µ )Z) t, (n) = 1 n n 7 and a slight modification of the FCLT, with no significant difference in proof, implies that {X;n)} and, therefore, {X;n°} converges in distribution to a Brownian motion with drift µ/Q and diffusion coefficient of I that starts at the origin. Let {X'} be a Brownian motion with drift It and diffusion coefficient a 2 starting at x. Then {W = (X, — x)/Q} is a Brownian motion with drift p/a and diffusion coefficient of! that starts at the origin. Hence, by using relation (3.8) of Section 3, P(i, <x) = P({X, } reaches c before d) = Pl{W} reaches c — x before d — x^ Q a = lim ({Sm,n : m = 0, 1, 2, ...} reaches cn before dn ) n-• ro d=x 1 — (ihn /^]n) a f = lim / d-x - c=x 1 — (pn/qn) a ✓n a ✓n µ d=x 1 + a n Q.J 1- 1— 7 = um n-ml^ d—c J 1 + a n U n 1— I I — ; µ^
- 46. FIRST PASSAGE TIME DISTRIBUTIONS FOR BROWNIAN MOTION 27 - exp c d ‚z x µ exp-2 A µ^ exp^(d —c) Ja 2 )j exp — d— - 1L vz Therefore, P(i' < zcd) = 1 — exp{2(d — x)p/vz } (c < x < d, p 0). (9.10) 1 — exp{2(d — c)p/v2 } If relation (3.6) of Section 3 is used instead of (3.8), then P(Td < T') = 1 — exp{ —2(x — c)µ/a2} (c <x < d, y 0). (9.11) 1 — exp{-2(d — c)µ/a} Letting d T oo in (9.10), one gets P(i<<oo)=exp{- 2(x z c)p } (c < x, p > 0), l o J)) (9.12) P(r <oo)= 1 (c<x,p<0). Thus, in this case the extremal random variable min,,, X° is exponentially distributed (Exercise 4). Letting c j — oc in (9.11) one obtains P(trd <oo)=1 (x<d,p>0), (9.13) P(-rd < oo) = exp{2(d — x)µ/a2 } (x < d, p < 0). In particular it follows that max,, o X° is exponentially distribute (Exercise 4). Relations (9.12), (9.13) imply that a Brownian motion with a nonzero drift is transient. This can also be deduced by an appeal to (a continuous time version of) the strong law of large numbers, just as in (3.11), (3.12) of Section 3 (Exercise 1). 10 FIRST PASSAGE TIME DISTRIBUTIONS FOR BROWNIAN MOTION We have seen in Section 4, relation (4.7), that for a simple symmetric random walk starting at zero, the first passage time 7y to the state y 0 0 has the
- 47. 28 RANDOM WALK AND BROWNIAN MOTION distribution N P(7.=N)=IYI N+y 1 N Y 2 N=IYI>IYI+2,IYI+4..... (10.1) Now let r = T° be the first time a standard Brownian motion starting at the origin reaches z. Let {X^") } be the polygonal process corresponding to the simple symmetric random walk. Considering the first time {X ) } reaches z, one has by the FLCT (Theorem 8.1) and Eq. 10.1 (Exercise 1), P(a= > t) = lim P(TZ f] > [nt]) n- X = lim P(T=,n] = N) n-+m N=(nt]+1 N = lim IYI ( N+ y N (Y = [z^]) n-+ao N=tnt]+1, N 2 N—yeven (10.2) Now according to Stirling's formula, for large integers M, we can write M! = (21r) 2e -MMnr+2 (1 + SM ) (10.3) where 8M —► 0 as M —► oo. Since y = [z], N> [nt], and 2(N ± y) > {[nt] — I[z/]I}/2, both N and N + y tend to infinity as n —• oo. Therefore, for N + y even, Ne-NNN+#2-N IYI N + y 2 _ N =IYI 2 N2 (2ir) t N e -(N+Y)12(N + Y) (N+y)/2+Ie —(N—Y)/2 (N — Y l (N-Yu2+# 2 ` 2 J X (1 + o(1)) (2ir)1I2N312 1+ N I 1— N (1 + o( 1 )) I (N + Y)/ 2 (N — Y)l2 (2ir) 2 /2N3/2 (1 + ) 1 — (1 + o( 1 )), N (10.4) where o(1) denotes a quantity whose magnitude is bounded above by a quantity en (t, z) that depends only on n, t, z and which goes to zero as n —• oo. Also,
- 48. FIRST PASSAGE TIME DISTRIBUTIONS FOR BROWNIAN MOTION 29 r y (N+yuz y wN-vuz _ N + Y Y _ YzIYI3 log [ (1 + N) 1 - N 2 N 2N2 +ß(N3) +N 2 y [ N+2N +O INI3/] 2 3 _ -2N+8(N,y), (10.5) where IB(N, y)j < n -11 z c(t, z) and c(t, z) is a constant depending only on t and z. Combining (10.4) and (10.5), we have N ( ^NN + Y 2-N =nN3I/2exp 1- z 2N}(1 + o( 1 )) 2 ( = n I N312 exp1-2N}(1 + 0(1)), (10.6) where o(1) --, 0 as n -* oo, uniformly for N> [nt], N - [z] even. Using this in (10.2), one obtains ( z P(r= > t) = lim rz IN312 expj -2N}. (10.7) n—^ao N> n:J, l N—[z.n) even Now either [nt] + 1 or [nt] + 2 is a possible value of N. In the first case the sum in (10.7) is over values N = [nt] - I + 2r (r = 1, 2, ...), and in the second case over values [nt] + 2r (r = 1, 2, ...). Since the differences in corresponding values of N/n are 2/n, one may take N = [nt] + 2r for the purpose of calculating (10.7). Thus, P(t2 > t) = lim ] + 2ex ( nz2 } fl X) n ([nt] + 2r)31^ p 1 2([nt] + 2r) _-Izl lim ^ 1 21 exp^- n r1 2 nzz 1 -.. = (t + 2r/n)3122(t + 2r/n) _ j2r fl - z2^ Izl 2 u3/^ exp - 2u du. (10.8) Now, by the change of variables v = Izi/ f , we get 2 folz P(T= > t) _ v e - "Z/z dv. (10.9) ^
- 49. 30 RANDOM WALK AND BROWNIAN MOTION The first passage time distribution for the more general case of Brownian motion {X1} with zero drift and diffusion coefficient Qz > 0, starting at the origin, is now obtained by applying (10.9) to the standard Brownian motion {(1/Q)X}. Therefore, 2 fI=Ibf P(;> t)_ e-°2/2 dv. (10.10) o The probability density function fQ2(t) of; is obtained from (10.10) as f 2(t) = Izle-ZZna=t (t > 0). (10.11) (2nci2 )1/2 t3/2 Note that for large t the tail of the p.d.f. f 2 (t) is of the order of t -3/2 . Therefore, although {X°} will reach z in a finite time with probability 1, the expected time is infinite (Exercise 11). Consider now a Brownian motion {X,} with a nonzero drift µ and diffusion coefficient a2 that starts at the origin. As in Section 9, the polygonal process {X^n)} corresponding to the simple random walk S,„,n = Z1 ,ß + • • • + Z„,,n, S0,, = 0, with P(Ztn,n = 1) = p„ = 2 + µ/(2Q), converges in distribution to {W = Xja}, which is a Brownian motion with drift µ/u and diffusion coefficient 1. On the other hand, writing T y,n for the first passage time of {S„, n : m = 0, 1, ...} to y, one has, by relation (4.6) of Section 4, N N P(1 = IY) p(N+v)/2R(N-v)/2 N) = N N + y 2 N IYI N ) (N-i-y)12( l (N-Y)l2 N+y 2 - 1+ 2— N Ql Q n FYI µ2 N/21 µ y/2 p y/2 = N N + y 12_N( a2 1 n 1 + / l — 2U n ; ) (10.12) For y = [w..J] for some given nonzero w, and N = [nt] + 2r for some given t> 0 and all positive integers r, one has ( _ N/2 / I1 1 } ^ Jv/z/l — ^ l-y/2 / µz +r J µ Wf/z µ w,,/„/z a2n )nt/2 1 + a nI l Q^) (1 + 0(1))
- 50. FIRST PASSAGE TIME DISTRIBUTIONS FOR BROWNIAN MOTION 31 = ex 2 2 r tµex ^w ex 1 l + 0 exp{ _ 4}( i — a2 n^ p 26 p 2v ( ()) t/1 2 /LW µz)n]rin (=exp--2+6 -zn1 +o(1)) r i ( z l ro exp — t µ 2 + ^W exp —Y- 2 + e„ J (l + 0(1)) (10.13) 2a Cr l Q where E„ does not depend on r and goes to zero as n —+ oc, and o(l) represents a term that goes to zero uniformly for all r >, 1 as n --* x. The first passage time i2 to z for {X1 } is the same as the first passage time to w = z/a for the process {I4'} = {X/a}. It follows from (9.12), (9.13) that if (i) p. <0 and z > O or (ii) p > 0 and z < 0, then there is a positive probability that the process { W} will never reach w = z/a (i.e., tZ = co). On the other hand, the sum of the probabilities in (10.12) over N> [nt] only gives the probability that the random walk reaches [w,,/n -] in a finite time greater than [nt]. By the FCLT, (10.6)—(10.8) and (10.12) and (10.13), we have 2 P(t < r < co) = IwI exp — tu2 + w µ lim 1 z n 2a Q „^^ r _ 1 n(t + 2r/n)312 wz x exp — + + e )'^” 2(t 2r/n) ^ tµ2 jJ 11 f w2 _ ^ (wI exp — 2a2 + 2 V312 exp — 2v x [ expf—^Zn`„-`I/2 du = I I p ^ —t1iW/.1 1 (2n)i/z w ex 2a a v3/z W2 ---- 2 x exp{— — (v — t) dv 2v 2az z z = w^ exp ^ 1 W/I) 1 exp — W — v A, (2n)l/Z a fj ^ v3'22v 2U2 for w = z/a. (10.14)
- 51. 32 RANDOM WALK AND BROWNIAN MOTION Therefore, for t > 0, z z Pt = 1i 1ex z -- v dv. 10.15 ( ) (27r) / Z a o2 v3/2 p2a2o 2QZ ( ) Differentiating this with respect to t (and changing the sign) the probability density function of rZ is given by .fa=. (t) = Izl exp (2 2o ^µz z2 — µZ t} nc2 )l / 2t3J2QZ 2QZt 2 Therefore, Izi .f«2 „(t) _ (2na2)1J2t3n exp{ --1 (z — µt)2 } (t > 0). (10.16) In particular, letting p(t; 0, y) denote the p.d.f. (7.1) of the distribution of the position X° at time t, (10.16) can be expressed as (see 4.6) .%2.u(t) = ICI p(t; 0, Z). (10.17) As mentioned before, the integral of J 2 ,2 (t) is less than 1 if either (i) p>O,z<Oor (ii) p<0,z>0. In all other cases, (10.16) is a proper probability density function. By putting p = 0 in (10.16), one gets (10.11). 11 THE ARCSINE LAW Consider a simple symmetric random walk {S,„} starting at zero. The problem is to calculate the distribution of the last visit to zero by So , S,, ... , S. For this we first calculate the probability that the number of + l's exceeds the number of — l's until time N and with a given positive value of the excess at time N. Lemma 1. Let a, b be two integers, 0 < a < b. Then P(S1>0,S2>0,...,S.+b-i> 0,Sa+b=b—a) b - a [(a+ b b — 11 — (a+b— 111^21a+n—(a b b)a+b(2)a+n. (11.1)
- 52. THE ARCSINE LAW 33 Proof. Each of the (t b) b) paths from (0, 0) to (a + b, b — a) has probability (2)° +b We seek the number M of those for which S 1 = 1, S2 > 0, S3 > 0, ... , Sa+b_1 > 0, Sa+b = b — a. Now the paths from (1,1) to (a + b, b — a) that cross or touch zero (the horizontal axis) are in one—one correspondence with those that go from (1, — 1) to (a + b, b — a). This correspondence is set up by reflecting each path of the last type about zero (i.e., about the horizontal time axis) up to the first time after time zero that zero is reached, and leaving the path from then on unchanged. The reflected path leads from (1, 1) to (a + b, b — a) and crosses or touches zero. Conversely, any path leading from (1,1) to (a + b, b — a) that crosses or touches zero, when reflected in the same manner, yields a path from (1, —1) to (a + b, b — a). But the number of all paths from (1, —1) to (a + b, b — a) is simply ("1') since it requires b plus l's and a — 1 minus l's among a + b — I steps to go from (1, —1) to (a + b, b — a). Hence M= (a b +b i1) — (a+b-1^ since there are altogether (' + 6 1 1 ) paths from (1,1) to (a + b, b — a). Now a straightforward simplification yields _ a+b b—a Mb )a+b Lemma 2. For the simple symmetric random walk starting at zero we have, f'(S154 0,S2^` 0,...,Sz" 0)=P(Sz"=0)_nn2n. (11.2) Proof. By symmetry, the leftmost side of (11.2) equals 2P(SI >0,S2 >0,...,S2n >0) " =2 Z P(S1 >O,S2 >0,...,SZ"_ 2 >0,Sen =2r) r=1 =2,= [ (n 2 +r 1 1) — (2n+r/](2) a" = 2(2nn 1 )2/ 2n = ( 2n)(^)'" n = P(S2" = 0), where we have adopted the convention that (2 2 n 1 ) = 0 in writing the middle equality.
- 53. 34 RANDOM WALK AND BROWNIAN MOTION Theorem 11.1. Let I'(') = max{ j: 0 ,<j ,<m, Si = 0}. Then P(F (2n) = 2k) = P(S2k = 0)P(S2n-2k = 0) = 2 k /2/ 2k(fl_kk2 ) J2n-2k (2k)!(2n — 2k)! (i'" = (k!)2 ((n — k)!)2 fork =0,1,2,..., n. (11.3) 2 Proof. By considering the conditional probability given {S2k = 0} one can easily justify that P(r(2n) = 2k) = P(S2k = 0, S2k +1 5 0, Sek + 2 0, ... , Sen * 0) = P(S2k = 0)P(SI0, S2 0, ... , S2n-2k 0) = P(S2k = 0)P(S2n-2k = 0)• Theorem 11.1 has the following symmetry relation as a corollary. P(r'(2n) = 2k) = P(17(2n) = 2n — 2k) for all k = 0,1, ... , n. (11.4) Theorem 11.2. (The Arc Sine Law). Let {B,} be a standard Brownian motion at zero. Let y = sup{t: 0 < t 5 1, B, = 0}. Then y has the probability density function 10<x<l. (11.5) i(x) _ IZ(x(1 — x))112 , P(y < x) = .f(y) dy = sin 1 x-. (11.6) fox n Proof. Let {So = 0, S1 , S2, ...} be a simple symmetric random walk. Define {X ' )} as in (8.4). By the FCLT (Theorem 8.1) one has P(y '< x) = um P(y(") < x) (0 < x < 1), n—ao where y(") = sup{t: 0 '< t '< 1, X = 0} = 1 sup{m: 0 < m <,n, S. = O} = 1 r("). n n In particular, taking the limit over even integers, it follows that P(y x) = limP (2n 1 r 2n) x I = um P(r(2nj < 2nx), n- 0, jjj n —ao
- 54. THE BROWNIAN BRIDGE 35 where I'(2n1 is defined in Theorem 11.1. By Theorem 11.1 and Stirling's approximation lim P(I'(Z"1 < 2nx) = lim [^1 (2k)!(2n — 2k)! 2_2n n-w n-•. k=o (k!)2 ((n — k)!)z 1-1 (27r) 112e - 2k(2k)zk+ j = lim Y- 112 -k k+})2 n-.00 k=o ((2n) e k (2x)1/2e-2(n-k)(2(n — k))[2cn-k)+#12-2n x ((2n)'1'e-("-k)(n — k)n-k++)2 [nxxl = lim Y- n-+oo k=o 7i (n — k) 1/2 1 Intl 11_ 1x 1 = n li k= n k k 112 n o (y(1 — y)) 1I2 dy • n^ 1 n// The following (invariance) corollary follows by applying the FCLT. Corollary 11.3. Let {Z,, Z2, ...} be a sequence of i.i.d. random variables such that EZ, = 0, EZi = 1. Then, defining {X^"1} as in (8.4) and y(n1 as above, one has lim P(y (n) x) = 2 sin - ' ^. (11.7) n-^M n From the arcsine law of the time of the last visit to zero it is also possible to get the distribution of the length of time in [0, 1] the standard Brownian motion spends on the positive side of the origin (i.e., an occupation time law) again.as an arcsine distribution. This fact is recorded in the following corollary (Exercise 2). Corollary 11.4. Let U = I {t < 1: Bt e IT + }I, where I I denotes Lebesgue measure and {B1} is standard Brownian motion starting at 0. Then, P(U < x) = 2 sin -1 ‚/i(11.8) n 12 THE BROWNIAN BRIDGE Let {B,} be a standard Brownian motion starting at zero. Since B, — tBl vanishes for t = 0 and t = 1, the stochastic process {B*} defined by
- 55. Other documents randomly have different content
- 56. right, and it had not been easy. She had not been surprised by his patient silence while she had been talking; for she had felt that it was hers to speak and his to listen. 'Thank you,' she said now. 'I shall never go back to what I have said, and neither of us need ever allude to old times again during this trip. It will not last long, for I shall probably go home by land from the first port we touch, and it is not likely that we shall ever meet again. If we do, I shall behave as if you were Count Kralinsky whom I have met abroad, neither more nor less. I suppose you will have conscience enough not to marry. Perhaps, if I thought another woman's happiness depended on it, I would consent to divorce you, but you shall never divorce me.' 'No power could make me wish to,' Kralinsky answered, still deeply moved. 'I was mad in those days, Maud; I was beside myself, between my debts and my entanglements with women not fit to touch your shoes. I've seen it all since. That is the chief reason why I chose to disappear from society when I had the chance, and become some one else! I swear to you, on my mother's soul in heaven, that I thought of nothing but that—to set you free and begin life over again as another man. No thought of marrying has ever crossed my mind! Do you think I could be as bad as that? But I'm not defending myself—how could I? All the right is on your side, and all the wrong on mine. And now—I would give heaven and earth to undo it all and to come back to you!' Lady Maud drew as far as she could into the corner where the fan- house joined the engine skylight. She had not expected this; it was too much repentance; it was too like a real attempt to win her again. He had not seen her for more than three months; she knew she was very beautiful; his fleeting passion had come to life again, as he had. But her old repulsion for him was ten times stronger than when they had parted, and she shrank back as far as she could, without speaking. From far below the noiseless engines sent a quick vibration up to the ironwork of the skylight. She felt it, but could
- 57. hardly tell it from the beatings of her own heart. He saw her shrinking from him and was wise. 'Don't be afraid of me!' he cried, in a low and pleading tone. 'Not that! Oh, please not that! I will not come nearer; I will not put out my hand to touch yours, I swear it to you! But I love you as I never loved you before; I never knew how beautiful you were till I had lost you, and now that I have found you again you are a thousand times more beautiful than in my dreams! No, I ask nothing! I have no right to ask for what I have thrown away! You do not even pity me, I think! Why should you? You were free when you thought me dead, and I have come back to be a burden and a weight on your life. Forgive me, forgive me, my lost darling, for the sake of all that might have been, but don't fear me! Pity me, if you can, but don't be afraid of me! Say that you pity me a little, and I shall be satisfied, and grateful too!' Lady Maud was silent for a few seconds, while he stood turned towards her, his hands clasped in a dramatic gesture, as if still imploring her commiseration. 'I do pity you,' she said at last, quite steadily, for just then she did not fear that he would try to touch even her hand. 'I pity you, if you are really in love with me again. I pity you still more if this is a passing thing that has taken hold of you merely because you still think me handsome. But I will never take you back to be my husband again. Never. That is finished, for good and all.' 'Ah, Maud, listen to me——' But she had already slipped out of the corner and was walking slowly away from him, not towards the others, but aft, so that he might join her quietly before going back to them. He was a man of the world and understood her, and did what was expected of him. Almost as soon as he was beside her, she turned to go forward with her leisurely, careless grace.
- 58. 'We've been standing a long time,' said she, as if the conversation had been about the weather. 'I want to sit down.' 'I am in earnest,' he said, very low. 'So am I,' answered Lady Maud. They went on towards the wheel-house side by side, without haste, and not very near together, like two ordinary acquaintances.
- 59. CHAPTER XIV While the Lancashire Lass was racing down to the Straits of Messina the Erinna was heading for the same point from the opposite direction, no longer dawdling along at half-speed, but going her full sixteen knots, after coaling in Naples, and any navigator who knew the positions and respective speeds of the two yachts could have calculated with approximate precision the point at which they would probably sight each other. Logotheti had given up the idea of taking Baraka to Paris, if he had ever really entertained it at all. He assured her that Naples was a great city, too, and that there was a first-rate French dressmaking establishment there, and that the Ville de Lyon would turn her out almost as smartly as the Rue de la Paix itself. He took Baraka ashore and placed her for half a day in the hands of Madame Anna, who undertook to do all that money could do in about a fortnight. He had the effrontery to say that Baraka was a niece of his from Constantinople, whose mother was on board the yacht, but had unfortunately sprained her ankle in falling down the companion during a gale, and could therefore not accompany her daughter on shore. The young lady, he said, spoke only Turkish. Madame Anna, grave and magnificently calm under all circumstances, had a vague recollection of having seen the handsome Oriental gentleman already with another niece, who spoke only French; but that was none of her business. When would the young lady try on the things? On any day Madame Anna chose to name; but in the meantime her uncle would take her down to Sicily, as the weather was so wonderfully fine and it was still so hot. Madame Anna therefore named a day, and promised, moreover, to see the best linen-drapers and sempstresses herself, and to provide the young lady with as complete an outfit as if she were going to be married. She should have all things visible and invisible in the shortest possible time.
- 60. Logotheti, who considered himself a stranger, insisted on putting down a thousand-franc note merely as a guarantee of good faith. The dressmaker protested almost furiously and took the money, still protesting. So that was settled, and Baraka was to be outwardly changed into a beautiful Feringhi lady without delay. To tell the truth, the establishment is really a smart one, and she was favourably impressed by the many pretty frocks and gowns that were tried on several pretty young women in order that she might make her choice. Baraka would have liked a blue satin skirt with a yellow train and a bright-green silk body, but in her travels she had noticed that the taste of Feringhi ladies was for very sober or gentle colours, compared with the fashionable standards of Samarkand, Tiflis, and Constantinople, and she meekly acquiesced to everything that Logotheti and Madame Anna proposed, after putting their heads together. Logotheti seemed to know a great deal about it. He took Baraka for a long drive in the afternoon, out by Pozzuoli to Baia and back. The girl loved the sea; it was the only thing in the western world that looked big to her, and she laughed at wretched little mountains only four or five thousand feet high, for she had dwelt at the feet of the lofty Altai and had sojourned in Tiflis under the mighty peak of Kasbek. But the sea was always the sea, and to her mountain sight it was always a new wonder beyond measure, vast, moving, alive. She gazed out with wide eyes at the purpled bay, streaked by winding currents of silver, and crisped here and there by the failing summer breeze. Logotheti saw her delight, and musical lines came back to him out of his reading, how the ocean is ever the ocean, and the things of the sea are the sea's; but he knew that he could not turn Greek verse into Turkish, try as he might, much less into that primeval, rough-hewn form of it which was Baraka's native tongue. It was nearly dark when the naphtha launch took them out to the yacht, which lay under the mole where the big English and German passenger steamers and the men-of-war are moored.
- 61. Logotheti had at last received Margaret's telegram asking him to meet her at once. It had failed to reach him in Gibraltar, and had been telegraphed on thence to Naples, and when he read it he was considerably disturbed. He wrote a long message of explanations and excuses, and sent it to the Primadonna at Bayreuth, tripling the number of words she had prepaid for his answer. But no reply came, for Margaret was herself at sea and nothing could reach her. He sent one of his own men from the yacht to spend the day at the telegraph office, with instructions for finding him if any message came. The man found him three times, and brought three telegrams; and each time as he tore open the little folded brown paper he felt more uncomfortable, but he was relieved to find each time that the message was only a business one from London or Paris, giving him the latest confidential news about a Government loan in which he was largely interested. When he reached the yacht he sent another man to wait till midnight at the office. The Diva was angry, he thought; that was clear, and perhaps she had some right to be. The tone of her telegram had been peremptory in the extreme, and now that he had answered it after a delay of several days, she refused to take any notice of him. It was not possible that such a personage as she was should have left Bayreuth without leaving clear instructions for sending on any telegrams that might come after she left. At this time of year, as he knew, she was beset with offers of engagements to sing, and they had to be answered. From eight o'clock in the morning to midnight there were sixteen hours, ample time for a retransmitted message to reach her anywhere in Europe and to be answered. Logotheti felt a sensation of deep relief when the man came aboard at a quarter- past midnight and reported himself empty-handed; but he resolved to wait till the following evening before definitely leaving Naples for the ten days which must elapse before Baraka could try on her beautiful Feringhi clothes. He told her anything he liked, and she believed him, or was indifferent; for the idea that she must be as well dressed as any
- 62. European woman when she met the man she was seeking had appealed strongly to her, and the sight of the pretty things at Madame Anna's had made her ashamed of her simple little ready- made serges and blouses. Logotheti assured her that Kralinsky was within easy reach, and showed no inclination to travel far. There was news of him in the telegrams received that day, the Greek said. Spies were about him and were watching him for her, and so far he had shown no inclination to admire any Feringhi beauty. Baraka accepted all these inventions without doubting their veracity. In her eyes Logotheti was a great man, something like a king, and vastly more than a Tartar chieftain. He could send men to the ends of the earth if he chose. Now that he was sure of where Kralinsky was, he could no doubt have him seized secretly and brought to her, if she desired it earnestly of him. But she did not wish to see the man, free or a prisoner, till she had her beautiful new clothes. Then he should look upon her, and judge whether he had done well to despise her love, and to leave her to be done to death by her own people and her body left to the vulture that had waited so long on a jutting point of rock over her head three years ago. Meanwhile, also, there were good things in life; there were very fat quails and marvellous muscatel grapes, and such fish as she had never eaten in Europe during her travels, and there was the real coffee of the Sheikhs, and an unlimited supply of rose-leaf preserve. Her friend was a king, and she was treated like a queen on the yacht. Every day, when Gula had rubbed her small feet quite dry after the luxurious bath, Gula kissed them and said they were like little tame white mice. Saving her one preoccupation, Baraka was in an Eastern paradise, where all things were perfect, and Kêf descended upon her every day after luncheon. Even the thought of the future was brighter now, for though she never left her cabin without her long bodkin, she was quite sure that she should never need it. In imagination she saw herself even more beautifully arrayed in Feringhi clothes than the pretty ladies with champagne hair whom she had seen driving in the Bois de Boulogne not long
- 63. ago when she walked there with Spiro. She wondered why Logotheti and Gula were both so much opposed to her dyeing her hair or wearing a wig. They told her that ladies with champagne hair were not always good ladies; but what did that matter? She thought them pretty. But she wondered gravely how Gula knew that they were not good. Gula knew a great many things. Besides, Baraka was 'good' herself, and was extremely well aware of the fact, and of its intrinsic value, if not of its moral importance. If she had crossed a quarter of the world in spite of dangers and obstacles which no European girl could pass unharmed, if alive at all, it was not to offer a stained flower to the man she sought when she found him at last. As for Logotheti, though he was not a Musulman, and not even an Asiatic, she felt herself safe with him, and trusted him as she would certainly not have trusted Van Torp, or any other European she had chanced to meet in the course of selling precious stones. He was more like one of her own people than the Greeks and Armenians of Constantinople or even the Georgians of the Caucasus. She was not wrong in that, either. Logotheti was beginning to wonder what he should do with her, and was vaguely surprised to find that he did not like the idea of parting with her at all; but beyond that he had no more thought of harming her than if she had been confided to his care and keeping by his own mother. Few Latins, whether Italians, French, or Spanish, could comprehend that, and most of them would think Logotheti a milksop and a sentimental fool. Many northern men, on the other hand, will think he did right, but would prefer not to be placed in such a trying position, for their own part, because beauty is beauty and human nature is weak, and the most exasperating difficulty in which an honest northern man can find himself where a woman is concerned is that dilemma of which honour and temptation are the two horns. But the best sort of Orientals look on these things differently, even when they are young, and their own women are safer with them
- 64. than European women generally are among European men. I think that most men who have really known the East will agree with me in this opinion. And besides, this is fiction, even though it be founded on facts; and fiction is an art; and the end and aim of art is always to discover and present some relation between the true and the beautiful—as perhaps the aim of all religions has been to show men the possible connexion between earth and heaven. Nothing is so easily misunderstood and misapplied as bare truth without comment, most especially when it is an ugly truth about the worst side of humanity. We know that all men are not mere animals; for heaven's sake let us believe that very few, if any, must be! Even Demopithekos, the mob- monkey, may have a conscience, when he is not haranguing the people. Logotheti certainly had one, of its kind, though he seemed to Margaret Donne and Lady Maud to be behaving in such an outrageous manner as to have forfeited all claim to the Diva's hand; and Baraka, who was a natural young woman, though a remarkably gifted and courageous one, felt instinctively that she was safe with him, and that she would not need to draw out her sharp bodkin in order to make her position clear, as she had been obliged to do at least twice already during her travels. Yet it was a dreamy and sense-compelling life that she led on the yacht, surrounded with every luxury she had ever heard of, and constantly waited on by the only clever man she had ever really talked with, excepting the old Persian merchant in Stamboul. The vision of the golden-bearded giant who had left her to her fate after treating her with stony indifference was still before her, but the reality was nearer in the shape of a visible 'great man,' who could do anything he chose, who caused her to be treated like a queen, and who was undeniably handsome. She wondered whether he had a wife. Judging marriage from her point of view, there probably had been one put away in that
- 65. beautiful house in Paris. He was an Oriental, she told herself, and he would not parade his wife as the Feringhis did. But she was one, too, and she considered that it would be an insult to ask him about such things. Spiro knew, no doubt, but she could not demean herself to inquire of a servant. Perhaps Gula had found out already, for the girl had a way of finding out whatever she wanted to know, apparently by explaining things to the second mate. Possibly Gula could be made to tell what she had learned, without being directly questioned. But after all, Baraka decided that it did not matter, since she meant to marry the fair-beard as soon as she had her pretty clothes. Yet she became conscious that if he had not existed, she would think it very satisfactory to marry the great man who could do anything he liked, though if he had a wife already, as he probably had, she would refuse to be the second in his house. The Koran allowed a man four, it was said, but the idea was hateful to her, and moreover the Persian merchant's wife had told her that it was old- fashioned to have more than one, mainly because living had grown so expensive. Logotheti sat beside her for hours under the awnings, talking or not, as she chose, and always reading when she was silent, though he often looked up to see if she wanted anything. He told her when they left Naples that he would show her beautiful islands and other sights, and the great fire-mountains of the South, Ætna and Stromboli, which she had heard of on her voyage to Marseilles but had not seen because the steamer had passed them at night. The fire-mountain at Naples had been quiet, only sending out thin wreaths of smoke, which Baraka insisted came from fires made by shepherds. 'Moreover,' she said, as they watched Vesuvius receding when they left Naples, 'your mountains are not mountains, but ant-hills, and I do not care for them. But your sea has the colours of many sherbets, rose-leaf and violet, and lemon and orange, and sometimes even of pale yellow peach-sherbet, which is good. Let me always see the sea till the fine dresses are ready to be tried on.'
- 66. 'This sea,' answered Logotheti, 'is always most beautiful near land and amongst islands, and the big fire-mountain of Sicily looks as tall as Kasbek, because it rises from the water's edge to the sky.' 'Then take me to it, and I will tell you, for my eyes have looked on the Altai, and I wish to see a real mountain again. After that we will go back and get the fine dresses. Will Gula know how to fasten the fine dresses at the back, do you think?' 'You shall have a woman who does, and who can talk with Gula, and the two will fasten the fine dresses for you.' Logotheti spoke with becoming gravity. 'Yes,' Baraka answered. 'Spend money for me, that I may be good to see. Also, I wish to have many servants. My father has a hundred, perhaps a thousand, but now I have only two, Gula and Spiro. The man I seek will think I am poor, and that will be a shame. While I was searching for him, it was different; and besides, you are teaching me how the rich Franks live in their world. It is not like ours. You know, for you are more like us, though you are a king here.' She spoke slowly and lazily, pausing between her phrases, and turning her eyes to him now and then without moving her head; and her talk amused him much more than that of European women, though it was so very simple, like that of a gifted child brought suddenly to a new country, or to see a fairy pantomime. 'Tell me,' he said after a time, 'if it were the portion of Kralinsky to be gathered to his fathers before you saw him, what would you do?' Baraka now turned not only her eyes to him but her face. 'Why do you ask me this? Is it because he is dead, and you are afraid to tell me?' 'He was alive this morning,' Logotheti answered, 'and he is a strong man. But the strong die sometimes suddenly, by accident if not of a fever.'
- 67. 'It is emptiness,' said Baraka, still looking at him. 'He will not die before I see him.' 'Allah forbid! But if such a thing happened, should you wish to go back to your own people? Or would you learn to speak the Frank and live in Europe?' 'If he were dead, which may Allah avert,' Baraka answered calmly, 'I think I would ask you to find me a husband.' 'Ah!' Logotheti could not repress the little exclamation of surprise. 'Yes. It is a shame for a woman not to be married. Am I an evil sight, or poor, that I should go down to the grave childless? Or is there any reproach upon me? Therefore I would ask you for a husband, because I have no other friend but only you among the Feringhis. But if you would not, I would go to Constantinople again, and to the Persian merchant's house, and I would say to his wife: "Get me a husband, for I am not a cripple, nor a monster, nor is there any reproach upon me, and why should I go childless?" Moreover, I would say to the merchant's wife: "Behold, I have great wealth, and I will have a rich husband, and one who is young and pleasing to me, and who will not take another wife; and if you bring me such a man, for whatsoever his riches may be, I will pay you five per cent."' Having made this remarkable statement of her intentions, Baraka was silent, expecting Logotheti to say something. What struck him was not the concluding sentence, for Asiatic match-makers and peace-makers are generally paid on some such basis, and the slim Tartar girl had proved long ago that she was a woman of business. What impressed Logotheti much more was what seemed the cool cynicism of her point of view. It was evidently not a romantic passion for Kralinsky that had brought her from beyond Turkestan to London and Paris; her view had been simpler and more practical; she had seen the man who suited her, she had told him so, and had given him the secret of great wealth, and in return she expected him
- 68. to marry her, if she found him alive. But if not, she would immediately take steps to obtain another to fill his place and be her husband, and she was willing to pay a high price to any one who could find one for her. Logotheti had half expected some such thing, but was not prepared for her extreme directness; still less had he thought of becoming the matrimonial agent who was to find a match worthy of her hand and fortune. She was sitting beside him in a little ready-made French dress, open at the throat, and only a bit of veil twisted round her hair, as any European woman might wear it; possibly it was her dress that made what she said sound strangely in his ears, though it would have struck him as natural enough if she had been muffled in a yashmak and ferajeh, on the deck of a Bosphorus ferry-boat. He said nothing in answer, and sat thinking the matter over. 'I could not offer to pay you five per cent,' she said after a time, 'because you are a king, but I could give you one of the fine rubies I have left, and you would look at it sometimes and rejoice because you had found Baraka a good husband.' Logotheti laughed low. She amused him exceedingly, and there were moments when he felt a new charm he had never known before. 'Why do you laugh?' Baraka asked, a little disturbed. 'I would give you a good ruby. A king may receive a good ruby as a gift, and not despise it. Why do you laugh at me? There came two German merchants to me in Paris to see my rubies, and when they had looked, they bought a good one, but not better than the one I would give you, and Spiro heard them say to each other in their own language that it was for their King, for Spiro understands all tongues. Then do you think that their King would not have been glad if I had given him the ruby as a gift? You cannot mock Baraka. Baraka knows what rubies are worth, and has some still.' 'I do not mock you,' Logotheti answered with perfect gravity. 'I laughed at my own thoughts. I said in my heart, "If Baraka asks me
- 69. for a husband, what will she say if I answer, Behold, I am the man, if you are satisfied!" This was my thought.' She was appeased at once, for she saw nothing extraordinary in his suggestion. She looked at him quietly and smiled, for she saw her chance. 'It is emptiness,' she said. 'I will have a man who has no other wife.' 'Precisely,' Logotheti answered, smiling. 'I never had one.' 'Now you are indeed mocking me!' she said, bending her sharp- drawn eyebrows. 'No. Every one knows it who knows me. In Europe, men do not always marry very young. It is not a fixed custom.' 'I have heard so,' Baraka answered, her anger subsiding, 'but it is very strange. If it be so, and if all things should happen as we said, which Allah avert, and if you desired me for your wife, I would marry you without doubt. You are a great man, and rich, and you are good to look at, as Saäd was. Also you are kind, but Saäd would probably have beaten me, for he beat every one, every day, and I should have gone back to my father's house. Truly,' she added, in a thoughtful tone, 'you would make a desirable husband for Baraka. But the man I seek must marry me if I find him alive, for I gave him the riches of the earth and he gave me nothing and departed, leaving me to die. I have told you, and you understand. Therefore let us not jest about these things any more. What will be, will be, and if he must die, it is his portion, and mine also, though it is a pity.' Thereupon the noble little features became very grave, and she leaned back in her chair and folded her hands in her lap, looking out at the violet light on the distant volcano. After that, at dinner and in the evening, they talked pleasantly. She told him tales of her own land, and of her childhood, with legends of the Altai, of genii and enchanted princesses; and he, in return, told her about the great
- 70. world in which he lived; but of the two, she talked the more, no doubt because he was not speaking his own language. Yet there was a bond of sympathy between them more natural and instinctive than any that had ever drawn him and Margaret together. When the sun was up the next morning and Logotheti came on deck to drink his coffee alone, he saw the magic Straits not many miles ahead, in an opalescent haze that sent up a vapour of pure gold to the pale blue enamel of the sky. He had been just where he was now more than once before, and few sights of nature had ever given him keener delight. On the left, the beautiful outline of the Calabrian hills descended softly into the still sea, on the right the mountains of Sicily reared their lofty crests; and far above them all, twice as high as the highest, and nobler in form than the greatest, Ætna towered to the very sky, and a vast cloud of smoke rose from the summit, and unfolded itself like a standard, in flowing draperies that streamed westward as far as the eye could reach. 'Let her go half-speed, Captain,' said Logotheti, as his sailing-master came up to bid him good-morning. 'I should like my guest to see the Straits.' 'Very good, sir. We shall not go through very fast in any case, for the tide is just turning against us.' 'Never mind,' Logotheti answered. 'The slower the better to-day, till we have Ætna well astern.' Now the tide in the Straits of Messina is as regular and easy to calculate as the tide in the Ocean, and at full and change of the moon the current runs six knots an hour, flowing or ebbing; it turns so suddenly that small freight steamers sometimes get into difficulties, and no sailing vessel I have ever seen has a chance of getting through against it unless the wind is both fresh and free. Furthermore, for the benefit of landsmen, it is well to explain here that when a steamer has the current ahead, her speed is the difference between her speed in slack water and that of the current
- 71. or tide, whereas, if the latter is with her, its speed increases her own. Consequently, though the Erinna could run sixteen knots, she would only be able to make ten against the tide; for it chanced that it was a spring tide, the moon being new on that very day. Similarly the Lancashire Lass, running her twenty-three knots like a torpedo boat, would only do seventeen under the same conditions.
- 72. CHAPTER XV At two o'clock in the morning Captain Brown was called by the officer of the watch, who told him that he was overhauling a good- sized steam yacht. The latter was heading up for the Straits from the southward, and the officer judged her to be not more than three or four miles on the port bow. Captain Brown, who meant business, was sleeping in his clothes in the chart-room, and was on the bridge in ten seconds, peering over the search-light with his big binocular. At two in the morning even the largest yachts do not show such a blaze of lights as passenger steamers generally do all night, and the one Captain Brown was watching had only two or three, besides the regulation ones. She might be white, too, though she might be a light grey, but he thought on the whole that she was painted white. She was rigged as a two-masted fore-and-aft schooner. So was the Erinna now, though she had once carried square topsails at the fore. She was also of about the same size, as far as it was possible to judge under the search-light. Captain Brown did not feel sure that he recognised her, but considering what his orders were he knew it was his duty to settle the question of her identity, which would be an easy matter in a quarter of an hour or less, as the course of the two vessels converged. He had been told to find the Erinna, but for what purpose he knew not, and he naturally supposed it to be a friendly one. As a first step, he ordered the Coston signal of his owner's yacht club to be burned, turned off the search-light, and waited for an answer. None came, however. Foreign yachts do not always burn signals to please vessels of other nations. A couple of minutes later, however, the white beam of a search-light shot out and enveloped Captain Brown and his ship. The other man
- 73. was evidently having a good look at him, for the light was kept full on for some time. But no signal was burned after it went out. Then Captain Brown turned on his own light again, and looked once more; and he had almost made up his mind that the other yacht was not quite as long as the Erinna, when she suddenly starboarded her helm, made a wide sweep away from him, and headed down the Sicilian coast in the direction of Catania. Captain Brown was so much surprised that he lowered his glasses and looked at his chief mate, whose watch it was, and who was standing beside him. It really looked very much as if the other vessel had recognised him and were running away. The chief mate also looked at him, but as they were more or less dazzled by the search- light that had been played on them, they could hardly see one another's faces at all. The captain wished his owner were on deck, instead of being sound asleep below. Owners who are not at all nautical characters do not like to be waked up at two o'clock in the morning by inquiries for instructions. Captain Brown considered the situation for two or three minutes before he made up his mind. He might be mistaken about the length and the bows of the Erinna, and if by any possibility it were she, he would not lose much by making sure of her. No other steamer could now pass out of the Straits without being seen by him. 'Hard-a-starboard,' he said to the mate. 'Hard-a-starboard,' said the mate to wheel. The big Lancashire Lass described a vast curve at her racing speed, while the captain kept his eye on the steamer he was going to chase. Before she was dead ahead the mate ordered the wheel amidships, and the Lancashire Lass did the rest herself. 'That will do for a course,' the captain said, when he had the vessel one point on the starboard bow. 'Keep her so,' said the mate to the wheel.
- 74. 'Keep her so, sir,' answered the quartermaster. It soon became clear to Captain Brown that he was chasing an uncommonly fast vessel, though he was willing to admit that he might have been a little out in judging the distance that separated him from her. Allowing that she might do sixteen knots, and even that is a high speed for yachts, he ought to have overtaken her in half an hour at the outside. But he did not, and he was much puzzled to find that he had gained very little on her when six bells were struck. Twice already he had given a little more starboard helm, and the pursued vessel was now right ahead, showing only her stern-light and the glare of her after-masthead light. 'Didn't I hear four bells go just after you called me?' he asked of the mate. 'Or was it five?' 'Four bells, sir. I logged it. At two-twenty we gave chase.' 'Mr. Johnson,' said the captain solemnly, 'he's doing at least twenty.' 'At least that.' The quartermaster who came to relieve the wheel at the hour, touched his cap, and reported eighty-five and eighty-six revolutions of the port and starboard engines respectively, which meant that the Lancashire Lass was doing her best. Then he took the other quartermaster's place. 'Chase,' said the man relieved. 'Keep her so.' 'Keep her so,' answered the other, taking over the wheel. Captain Brown spoke to his officer. 'Tell them to try and work the port engine up to eighty-six, Mr. Johnson.' The chief mate went to the engine telephone, delivered the message, and reported that the engineer of the watch in the port
- 75. engine said he would do his best, but that the port engine had not given quite such a good diagram as the starboard one that morning. Then something happened which surprised and annoyed Captain Brown; and if he had not been a religious man, and, moreover, in charge of a vessel which was so very high-class that she ranked as third in the world amongst steam yachts, and perhaps second, a fact which gave him a position requiring great dignity of bearing with his officers, he would certainly have said things. The chased vessel had put out her lights and disappeared into complete darkness under the Sicilian coast. Again he and his officer looked at one another, but neither spoke. They were outside the wheel-house on the bridge on the starboard side, behind a heavy plate-glass screen. The captain made one step to the right, the mate made one to the left, and both put up their glasses in the teeth of the gale made by the yacht's tremendous way. In less than a minute they stepped back into their places, and glanced at each other again. Now it occurred to Captain Brown that such a financier as his owner might be looking out for such another financier as the owner of the Erinna for some reason which would not please the latter, whose sailing-master had without doubts recognised the Lancashire Lass at once, because she was very differently built from most yachts. 'Search-light again, Mr. Johnson,' said the captain. The great beacon ran out instantly like a comet's tail, and he stood behind it with his glasses. Instead of a steamer, he saw a rocky islet sticking up sharp and clear, half a point on the starboard bow, about three miles off. It was the largest of the Isles of the Cyclops, as he very well knew, off Aci Reale, and it was perfectly evident that the chased vessel had first put out her lights and had then at once run behind the islands, close inshore. Captain Brown reflected that the captain he was after must know the waters well to do such a thing, and that the deep draught of his own ship made it the height of folly
- 76. to think of imitating such a trick at night. Yet so long as the other stayed where she was, she could not come out without showing herself under his search-light. 'Half-speed both engines,' he said quickly. The mate worked the engine telegraph almost as soon as the captain began to speak. 'Starboard five degrees more,' said Captain Brown. The order was repeated to the wheel, and the quartermaster gave it back, and repeated it a second time when the vessel's head had gone off to port exactly to the required degree. 'Slow,' said Captain Brown. 'Stop her,' he said a moment later. Twin-screw steamers cannot be stopped as quickly by reversing as those with a single screw can, and the Lancashire Lass would keep way on for three miles or more, by which time she would be abreast of the islands, and at a safe distance from them. Besides, the spring tide was now running fresh down the Straits, making a current along the coast, as Captain Brown knew. The instant the engines stopped, the third mate came round from the chart-room, where he had been sent to work a sight for longitude by Aldebaran for the good of his young nautical soul. A moment later Mr. Van Torp himself appeared on the bridge in pyjamas. 'Got her?' he asked eagerly. Captain Brown explained that he thought he had cornered the Erinna behind the islet, but was not quite sure of her. Mr. Van Torp waited and said nothing, and the chief mate kept the search-light steadily on the rocks. The yacht lost way rapidly, and lay quite still with the islet exactly abeam, half a mile off, as the captain had calculated. He then gave the order to go slow ahead.
- 77. A minute had not passed when the vessel that had lain concealed behind the island ran out suddenly with all her regulation lights up, apparently making directly across the bows of the Lancashire Lass. Now the rule of the road at sea requires every steamer under weigh to keep out of the way of any steamer that appears on her starboard side forward of the beam. At such a short distance Captain Brown had hardly any choice but to stop his ship again and order 'half- speed astern' till she had no way, and he did so. She was barely moving when the order was given, and a few turns of the engines stopped her altogether. 'Is that the Erinna, Captain?' asked Mr. Van Torp. Captain Brown had his glasses up and did not answer at once. After nearly a minute he laid them down on the lid of the small box fastened to the bridge-rail. 'No, sir,' he answered in a tone of considerable disappointment. 'At four miles' distance she looked so much like her that I didn't dare to let her slip through my fingers, but we have not lost more than a couple of hours.' 'What is this thing, anyway? She's coming towards us pretty quick.' 'She's one of those new fast twin-screw revenue cutters the Italians have lately built, sir. They look very like yachts at night. There's a deal of smuggling on this coast, over from Malta. She's coming alongside to ask what we mean by giving chase to a government vessel.' Captain Brown was right, and when the big cutter had crossed his bows, she ran all round him while she slowed down, and she stopped within speaking distance on his starboard side. The usual questions were asked and answered. 'English yacht Lancashire Lass, from Venice for Messina, expecting to meet a friend's yacht at sea. Thought the revenue cutter was she.
- 78. Regretted mistake. Had the captain of the cutter seen or heard of English yacht Erinna?' He had not. There was no harm done. It was his duty to watch all vessels. He wished Captain Brown a pleasant trip and good-night. The Italian officer spoke English well, and there was no trouble. Revenue cutters are very civil to all respectable yachts. 'Hard-a-starboard. Port engine slow astern, starboard engine half- speed ahead.' That was all Captain Brown said, but no one could guess what he was thinking as his big vessel turned quickly to port on her heel, and he headed her up for the Straits again. Mr. Van Torp said nothing at all, but his lips moved as he left the bridge and went off to his own quarters. It was now nearly four o'clock and the eastern sky was grey. The current was dead against the yacht through the Straits, which were, moreover, crowded with all sorts of large and small craft under sail, taking advantage of the tide to get through; many of them steered very badly under the circumstances, of course, and it was out of the question to run between them at full speed. The consequence was that it was eight o'clock when the Lancashire Lass steamed slowly into Messina and dropped anchor out in the middle of the harbour, to wait while Captain Brown got information about the Erinna, if there were any to be had at the harbourmaster's office. It would have been folly to run out of the Straits without at least looking in to see if she were there, lying quietly moored behind the fortress of San Salvatore and the very high mole. She was not there, and had not been heard of, but a Paris Herald was procured in which it was stated that the Erinna had arrived in Naples, 'owner and party on board.' 'Well,' said Mr. Van Torp, 'let's get to Naples, quick. How long will it take, Captain?'
- 79. 'About eight hours, sir, counting our getting under weigh and out of this crowded water, which won't take long, for the tide will soon turn.' 'Go ahead,' said Mr. Van Torp. Captain Brown prepared to get under weigh again as quickly as possible. The entrance to Messina harbour is narrow, and it was natural that, as he was in a hurry, a huge Italian man-of-war should enter the harbour at that very moment, with the solemn and safe deliberation which the movements of line-of-battle ships require when going in and out of port. There was nothing to be done but to wait patiently till the fairway was clear. It was not more than a quarter of an hour, but Captain Brown was in a hurry, and as there was a fresh morning breeze blowing across the harbour he could not even get his anchor up with safety before he was ready to start. The result of all these delays was that at about nine o'clock he saw the Erinna right ahead, bows on and only half a mile away, just between Scylla and Faro, where the whirlpool is still a danger to sailing vessels and slow steamers, and just as the tide was turning against her and in his own favour. He did not like to leave the bridge, even for a moment, and sent the second mate with an urgent message requesting Mr. Van Torp to come up as soon as he could. Five minutes earlier the owner had sat down to breakfast opposite Lady Maud, who was very pale and had dark shadows under her eyes for the first time since he had known her. As soon as the steward left them alone, she spoke. 'It is Leven,' she said, 'and he wants me to take him back.' Mr. Van Torp set down his tea untasted and stared at her. He was not often completely taken by surprise, but for once he was almost speechless. His lips did not even move silently. 'I was sure it was he,' Lady Maud said, 'but I did not expect that.' 'Well,' said Mr. Van Torp, finding his voice, 'he shan't. That's all.'
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