![Weak Convergence and Uniformity Classes
uniformity classes in Section 2. These two sections are not used in the
sequel (except as motivation). In Section 3 we obtain estimates such as
sup 4((aC))<d(k)e (e>0), (3)
CEC
where 4) is the standard normal distribution on R', C is the class of all
(Borel-measurable) convex subsets of R k, and d(k) is a positive number
depending only on k. We have several occasions in Chapters 3 and 4 to use
these estimates for deriving rates of convergence Q„(C)-->4'(C), C E C',
where Q„ is the distribution of the normalized sum of n independent
random vectors.
1. WEAK CONVERGENCE
In this section we briefly review some standard results in the theory of
weak convergence of probability measures.
Throughout this section S denotes a metric space with a metric p. The
Borel sigma field 3 of S is the smallest sigma-field containing the class of
all open subsets of S. We say is a (signed) measure on S if it is a (signed)
measure defined on . The class of all finite signed measures on S is
denoted by OiL, and the subclass of O1i, comprising all probability mea-
sures is denoted by 'P. Given a finite signed measure s on S, one defines
three associated set functions called the positive, negative, and
total variations of µ, respectively, by
(B)=sup {µ(A):ACB, AE),
(B)=
µ - (B)=— inf( (A):ACB, AE°.1J), (BEAU) (1.1)
1µ1=µ++ tL .
The so-called Jordan-Hahn decompositions asserts that µ+ and s - (and,
therefore I s) are finite measures on S satisfying
(1.2)
For every finite signed measure s on a separable metric space S, we define
the support of as the smallest closed subset of S whose complement has
I µI-measure zero: that is,
support of µ= n {F:F closed, IµI(S F)=0}, (1.3)
tSee Halmos [1], pp. 121-123.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-29-320.jpg)
![Weak Convergence 3
where for any two sets A, B we write
AB={x:xEA,x^B}. (1.4)
Note that the separability of the metric space S ensures that the comple-
ment of the right side of (1.3) has zero I SI-measure.
The class OIL of (set) functions on J into R 1 is a real linear space with
respect to pointwise addition and multiplication by real scalars. It is a
Banach space when endowed with the variation norm
IIµII=IIl(S) (ttEOIL). (1.5)
Let C(S) denote the class of all complex-valued, bounded, continuous
functions on S. The weak totopogy on Olt, is the weakest topology (on OIL)
that makes the maps
µ— f fdµ [f EC(S)] (1.6)
on cX into the complex field C continuous. The right side of (1.6) always
stands for the Lebesgue integral of (a µ-integrable, complex-valued, Borel-
measurable function) f on S. The Lebesgue integral of f on a Bore! set B is
denoted by
fBfdµ. (1.7)
When it becomes necessary to indicate the variable of integration, we also
write
f f (x) µ(dx) (1.8)
instead of f f dµ.
In this monograph we are particularly concerned with the relativized
weak topology on the class 6P of all probability measures on S. In this
topology convergence of a sequence (Q„) of probability measures to a
probability measure Q means
lira f fdQ„= f fdQ (1.9)
for every f in C(S). The following theorem gives several characterizations
of weak convergence of a sequence of probability measures.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-30-320.jpg)
![4 Weak Convergence and Uniformity Classes
THEOREM 1.1. Let S be a metric space. Let Q. (n= 1,2,... ), Q be
probability measures on S. The following statements are equivalent.
(i) Q„ converges weakly to Q.
(ii) lin
en f f dQ,, = ffdQ
dQ for every uniformly continuous f in C (S).
(iii) urn Q. (F) < Q (F) for every closed subset F of S.
(iv) lint Q„(G)> Q(G) for every open subset G of S.
n
For a proof of this theorem we refer to Billingsley [1] (Theorem 2.1, pp.
11-14) or Parthasarathy [1] (Theorem 6.1, pp. 40-42).
Let B (x : E) denote the open ball with center x and radius E,
B(x:E)={y:yES, p(x,y)<E} (xES, c>0). (1.10)
For an arbitrary real-valued function f on S we define, for each positive
number e, the oscillation function wj(-
wf(x:E)= sup {I f(z)-f(y)I:y,zEB(x:c)} (x ES). (1.11)
For a complex-valued function f= g + ih (g, h real-valued), define
Wf(X:E)— wg (X:E)+Wh(X:E) (xES, E >0). (1.12)
The oscillation function is Borel-measurable on the (Borel-measurable) set
on which it is finite.t The set of points of discontinuity of f is Borel-
measurable and can be expressed as
(x:wj (x: 1 ) +0 as n-moo}. (1.13)
n 1
Now let Q be a probability measure on S. A complex-valued function f on
S is said to be Q-continuous if its points of discontinuity comprise a set of
Q-measure zero. In particular, if the indicator function IA of a set A, taking
values one on A and zero on the complement of A, is Q-continuous, we say
A is a Q-continuity set. Since the set of points of discontinuity of IA is
precisely the boundary 8A of A, A is a Q-continuity set if and only if
Q(aA)=0. (1.14)
Recall that the (topological) boundary aA of a set A is defined by
aA =Cl(A)Int(A), (1.15)
tSee relations (11.1)—{11.4) and the discussion following them.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-31-320.jpg)
![6 Weak Convergence and Uniformity Classes
and
p(x,A)=inf (p(x,y):yEA}. (1.18)
For these and other details we refer the reader to Billingsley [1], (Appendix
III, pp. 233-242) or Parthasarathy [1] (Chapter II, pp. 39-55).
2. UNIFORMITY CLASSES
Unless otherwise specified, throughout this section S denotes a separable
metric space.
Let Q be a given probability measure on S. Our main task in this section
is to characterize those classes f of complex-valued, bounded, Borel-
measurable functions on S for which
lim sup f fdQ„— f fdQI =0 (2.1)
n fEc
for every sequence of probability measures (Q„) converging weakly to Q.
Such a class is called a Q-uniformity class of functions. A class 9 of
Borel subsets of S is called a Q-uniformity class of sets if
lim sup IQ(A)—Q(A)I=0(2.2)
n AE(^
for every sequence of probability measures {Q„) converging weakly to Q.
Thus (t' is a Q-uniformity class of sets if and only if = { IA : A E Q) is a
Q-uniformity class of functions. We need some preparation before we can
characterize uniformity classes.
In the following lemma we deal with (signed) measures on a sigma-field
of subsets of an abstract space I. For a finite signed measure on this
space the variation norm II µII is again defined by (1.5) with 2 replacing S
(and C replacing i3
( ).
LEMMA 2.1. (Scheffe's theorem) Let (St, LS ,A) be a measure space. Let
Q„ (n = 1,2,...), Q be probability measures on (S2, in ) that are absolutely
continuous with respect to A and have densities (i.e., Radon-Nikodym deriva-
tives) q„(n = 1,2,...), q, respectively, with respect to A. If (q„) converges to q
almost everywhere (A), then
linmIIQ,,—Q11=O.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-33-320.jpg)
![Uniformity Classes 11
NECESSITY. Assume (2.9i) does not hold. Then there exists a sequence
{ f,,) c such that
mJ (S)>n (n=1,2,...). (2.21)
Thus if we write
a„=inf(f„(x):xES}, Q=sup(f„(x):xES}, (2.22)
then /3„ — a,, > n for all n. Divide the closed interval [a,,, /3J into n disjoint
subintervals of equal length. There exists a subinterval I, such that
Q(f^ I (I.,))>n (n=1,2,...). (2.23)
Also, since /3,, — an > n, there exists a point x,, in S for which
If„(x,,)—tl. n 2 for all tECI(I,,) (n=1,2,...). (2.24)
Now define a probability measure Q„ by adding a point mass 1/n at x,,
and subtracting this mass proportionately from subsets of f,^ '(I„); that is,
Q.(A) = Q(Afn '(In))+ — (A)
+f 1— nQ( f^ ^(I^)) ]Q(Anf„ '(I.)), (AE), (2.25)
where Sx. is the probability measure degenerate at x„ (i.e., Sx. ({ x,, }) = 1).
Note that
IIQ.-Q11=n, (2.26)
so that Q,, converges in variation norm and, therefore, weakly to Q. But
f f dQ,, — Jf,,dQl = n II„(x,,)— Q f^
1^ (I^) ✓J (i^)l,,dQ](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-38-320.jpg)
![Uniformity Classes 15
so that in this case is a Q-uniformity class if and only if
lim sup Q((aA)`)=0. (2.41)
fio AE&
Proof. For any arbitrary set A, wrA (x : e) is one if and only if B (x : e)
intersects both A and SA; otherwise it has the value zero. Therefore
W]A (X:E)=,A 'A -.(x) (XES, E>0). (2.42)
Hence
fwjA (x:e)Q(dx)=Q(A`A - `) (ACS, e>0). (2.43)
Since w,A (S) < I for all sets A, it follows from Theorem 2.5 that (2.39) is a
necessary and sufficient condition for & to be a Q-uniformity class.
We now prove (2.40) under the hypothesis that every open ball of the
metric space S [separability is not needed for the validity of (2.40)] is
connected. First we show that the relation
(aA)`CA`A - ` (ACS, e>0) (2.44)
is valid in every metric space S. For suppose xE(aA)`. Then there exist a
positive c' smaller than and a pointy in aA such that p(x,y) < e'. Since y
is a boundary point of A, there exist two points z, and z2 , with z, in A and
z2 in S A, such that p(y, z.) < E — e' for i = 1, 2. Thus p(x, z.) < p(x,y) +
p(y, z; ) < e for i= 1, 2. This means that x E A ` and x A - `, which proves
(2.44). Next we assume that every open ball of S is connected. If x E A `
A -`, then
AnB(x:e)#4, (SA)nB(x:e)^. (2.45)
We now suppose that xE(8A) (and derive a contradition). This means
B(x':e)n 8A=0, (2.46)
so that
B(x:e)=((SCl(A))n B(x:E))u(Int(A)nB(x:e)), (2.47)
since S = (S C 1(A )) U Int (A) U aA
A. The right side of (2.47) is the disjoint
union of two open sets. These two sets are nonempty because of (2.45),
(2.46), and the relations (which hold in every topological space)
(SC1(A))U aA D SA, Int(A)U aA DA.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-42-320.jpg)
![18 Weak Convergence and Uniformity Classes
where 4, is defined on [0, oo) by
(t) __1. 1—t if 0 < t < 1,
(2.56)
0 if 1>1.
Note that f is, for every positive e, a bounded Lipschitzian function
satisfying
wf(S)< 1,If,(X) — f^(Y)I<) EP(x+F) — - p(Y,F)I
EP(x,Y) (x, Y E S ),
so that, by (2.53),
linm f f^ dQ„ = f f, dQ (e>0). (2.57)
Since IF f, for every positive e,
lin
m Q,, (F) < lim f f dQ„ = f f dQ (e>0).(2.58)
Also, lim f^(x)= IF(x) for all x in S. Hence
lim JJdQ=Q(F). (2.59)
cj0
By Theorem 1.1 { Q) converges weakly to Q. Finally, it is easy to check
that dBL is a distance function on 'P. Q.E.D.
Remark. The distance daL may be defined on the class GR, of all finite signed measures on S
by letting Qt . Q2 be finite signed measures in (2.51). The function µ—.d,L (s,0) is a norm on
the vector space 9R.. The topology induced by this norm is, in general, weaker than the one
induced by the variation norm (1.5). It should also be pointed out that the proof of Corollary
2.8 presupposes metrizability of the weak topology on 9 and merely provides a suitable
metric as an alternative to the Prokhorov distance do defined by (1.16)]. This justifies the use
of sequences (rather than nets) in the proof.
One can construct many interesting examples of uniformity classes
beyond those provided by Corollaries 2.7 and 2.8. We give one example
now. The rest of this section will be devoted to another example (Theorem
2.11) of considerable interest from our point of view.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-45-320.jpg)
![Uniformity Classes 19
Example Let S=R 2. Let 6! (1) be the class of all Borel-measurable
subsets of R 2 each having a boundary contained in some rectifiable curvet
of length not exceeding a given positive number 1. We now show that 6D (I)
is a Q-uniformity class for every probability measure Q that is absolutely
continuous with respect to the Lebesgue measure A 2 on R 2. Let A E 6 (l)
and let 3A C J, where J is a rectifiable curve of length 1. There exist k
points zo,z t ,.••,zk -, on J such that (i) zo and zk _ t are the end points of J
(may coincide), (ii) k<I/c+2, and (iii) Iiz1 —z1 _,II<E for i=1,2,...,k-1
(11.11 denotes euclidean norm). The k open balls having z;'s as centers and
radii 2E cover J`. Hence
A2((aA)`) <1A2(J (L +2)1T(2E)
2
=41rk+87rE 2[A E (l )]. (2.60)
Let Q be a probability measure that is absolutely continuous with respect
to A2. Then (2.60) implies (in view of Theorem A.3.1)
lim (sup{Q((aA):AE^'i)(1)})=0. (2.61)
By Corollary 2.6, -D (1) is a Q-uniformity class.
We need some preparation before proving the next theorem. Let S be a
metric space. Define the Hausdorff distance A between two closed bounded
subsets A, B of S by
A(A,B)=inf{ E:c>0,AcB`, BCA`}. (2.62)
The class Q of all closed bounded subsets of S is a metric space with
metric A.
LEMMA 2.9. Let ')1t be a compact subset of (? . Then for every probabil-
ity measure Q on S one has
lim sup{Q(M`):ME )=sup{Q(M):MElt). (2.63)
cjo
Proof Let rq be a given positive number. For every M E )lt. there exists a
positive number E,t, such that
Q(M`")<Q(M)+"1 ,
to rectifiable curve in R 2 is a subset of R 2 of the form {z(t):0G t < 1), where t- z(t)
=(x(t),y(t)) is a continuous function of bounded variation on [0,11 into R 2 ; z(0), z(1) are
called the endpoints of the curve.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-46-320.jpg)
![20 Weak Convergence and Uniformity Classes
since M`J.M as 40. By compactness of Olt, there exists a finite collection
of sets {M1 , ..., M,) in OL such that every set in OJlt, is within a A-distance
from M for some i, 1 <i < r. Let
EM;
Eo =-min { :1<i<r } .
Then
sup(Q(M`"):MEOL) <sup{Q(M,`°Mi2):1<i<r}
<sup(Q(MiAli ): 1 <i <r)
<sup {Q(Mi )+rl: 1 < i < r}
<sup (Q(M): M E Olt,}+rl.
Q.E.D.
A subset C of Rk is convex if a x, + (1 — a) x2 E C for all x,, x2 E C
and all a E [0,1]. A hyperplane H in R' is a set of the form
H={x:<u,x>=c}, (2.64)
where c is a real number and u is a unit vector of R k ; that is, Ilul) = 1, and
<,> denotes euclidean inner product
k
<u,X>= u,X,, (2.65)
i=!
k 1/2
(lull u? [u=(u1,...,uk), x=(x 1 ,...,xk )ER k ].
i^l
A closed half space E is a set of the form
E=(x:xER k ,<u,x)<c) {cER', uER k , IIull=l]. (2.66)
A hyperplane H= (y : <u,y> = c) is said to be a supporting hyperplane for a
set A at x E A if
<u,x>=c, AC{z:<u,z><c}, (2.67)
that is, if the hyperplane H passes through the point x of A and has A on
one side. Note that the sign < in (2.66) and (2.67) may be replaced by >](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-47-320.jpg)
![22 Weak Convergence and Uniformity Classes
for every c > 0(A, B) and every positive rt. Hence
0(aA,aB) <0(A,B). (2.70)
The inequalities (2.69) and (2.70) together yield (2.68). Q.E.D.
THEOREM 1.11. Let C denote the class of all Bore/-measurable convex
subsets of Rk. Let Q be a probability measure on R k. The class e is a
Q-uniformity class if and only if it is a Q-continuity class, that is, if and only
if
Q(ac)=o forallCEe. (2.71)
Proof If e is a Q-uniformity class, then, by relation (2.41) in Corollary
2.6, e is a Q-continuity class. Suppose, conversely, that (2.71) holds. Since
the characterization (2.41) of Q-uniformity is in terms of boundaries, and
since aC= a (Cl(C)) for every convex set [note that C1(C)= Int(C)u aC
and that aC has empty interior for convex C], it follows that C is a
Q-uniformity class if and only if e = (C 1(C) : CE C) is. Given ri >0, let r
be so chosen that
Q({x:IIxII>r))<2. (2.72)
Write
C', = { C: C E e, CCC1(B (0: r))).
By a well-known theorem of Blaschket 3, is compact in the Hausdorff
metric A defined by (2.62). Lemma 2.10 shows that the compactness of ,
is equivalent to the compactness of { aC : C E 13, }. Lemma 2.9 and the
hypothesis (2.71) now yield
lim sup{ Q((8C)`):CEC',}=0. (2.73)
By Corollary 2.6, C, is a Q-uniformity class. Let (Qn ) be a sequence of
probability measures weakly converging to Q. Then
lim (sup{IQn(C) — Q(C)I:CEC^))
n
<11im (sup{IQ.(C)—Q(C)^:CE,})
+ lim Qn({x:llxll>r))+Q({x:llxll>r))
n
=2Q({x:fIxll >r})<ii. (2.74)
tSee Eggleston [1), pp. 34-67.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-49-320.jpg)
![Integrals Over Convex Shells 23
Note that the last equality follows from the fact that (x : Ilxil > r) is a
Q-continuity set, since its complement is [although, given any probability
measure Q and a positive rl, one can always find r such that (x: jlxjl > r) is
a Q-continuity set and (2.72) holds]. Since rl is an arbitrary positive
number, it follows from (2.74) that a is a Q-uniformity class. Con-
sequently, (2 is a Q-uniformity class. Q.E.D.
Remark. It follows from the above theorem that if Q is absolutely continuous with respect
to Lebesgue measure on R", then e is a Q-uniformity class. In particular, a is a 4)-
uniformity class, 4) being the standard normal distribution in R k . We shall obtain a
refinement of this last statement in the next section.
It is easy to see from the proof of Theorem 2.11 that the class C in its statement may be
replaced by any class Ef of Borel-measurable convex sets with the property that
R,- (CI(A)nCl(B(0:r)):AE($}
is compact in the Hausdorff metric for all positive r. In particular, by letting
d — {(—oo,x1]X(—oo,x2]X ... X( — oo,xk]:x—(xi,...,Xk)ER k },
we get Polya's result: Let P be a probability measure on Rk whose distribution function F,
defined by
F(x)-P((-oo,x,]x... x(-oo,xk ]) [x-(x......xk)ERk ], (2.75)
is continuous on Rk. If a sequence of probability measures {P„} with distribution functions {FA )
converges weakly to P, then
sup(IF„(x)-F(x)I:xER k }-+0 (2.76)
as n-boo. The left side of (2.76) is sometimes called the Kolmogorov distance between Po and P.
The converse of this result is also true: if (2.76) holds, then (P„) converges weakly to P. In
fact, if (F„} converges to Fat all points of continuity of F, then (P„) converges weakly to P.t
3. INEQUALITIES FOR INTEGRALS OVER CONVEX SHELLS
It is not difficult to check that if C is convex then so are Int(C),C1(C).
The convex hull c(B) of a subset B of R k is the intersection of all convex
sets containing B. Clearly c(B) is convex. If C is a closed and bounded
convex subset of Rk, then it is the convex hull of its boundary; that is,
c(8C)=C.
Clearly c(8C) C C. On the other hand, if x E C, x li= 8C, then every line
through x intersects aC at two points and x is a convex combination of
these two points. Thus c(aC) = C. If C is convex and e >0, then C` is
tSee Billingsley [ 1 1, pp. 17-18.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-50-320.jpg)
![Betaught, [bi-tahte] == taught. Rel. S. v. 124
Bete, v. a. lit. == ‘make better;’ hence, ‘heal,’ ‘save.’ Marg. 68
—— == ‘recompense,’ ‘make amends for.’ RG. 369. AS. bétan
Bete, part. == beaten. Vid. Beat
Beten, [y-beten] == overlaid, covered, as with silk, gold, &c. Alys.
1034, 1518
Beth, Beoth, &c. See Be
Bethink, v. a. == ‘to bethink oneself’ of a thing. RG. 368, 458
Betide, v. n. == happen. RG. 418, 14
Betime, adv. K. Horn, 995
Betoken, v. a. RG. 152
Betokening, sb. RG. 560
Betray, v. a. RG. 135
Better, adj. RG. 367, 422
—— v. n. == get the better. Ps xii. 5
Betterness, sb. Ps. li. 5
Between, prep. RG. 371, 513
Betwixt, prep. [bi-tuxen]. O. and N. 1745
Beverage, sb. == drink. RG. 26
—— == reward, consequence. RG. 299
Bewail, v. a. Alys. 4395
Beware, v. n. RG. 547](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-56-320.jpg)
![Beweep, v. a. O. and N. 972
Bewind, v. a. == entwine. part. ‘bewound.’ Christ on the Cross, 3
Bewray, v. a. == betray [by-wrye]. Alys. 4377. pret. ‘bi-wro.’ O. and
N. 673. AS. wrégan.
Beyen, == are? Wright’s L. P. p. 32
Beyond, prep. RG. 368, 420
Beyre, == of both, gen. pl. RG. 388, 398
Bezant, sb. == a piece of money. RG. 409. From Byzantium, or
Constantinople, where they were originally used
Bible, sb. Rel. Ant. ii. p. 174
Bicast, v. a. == cast over, cover. 92 β
Bicatch, v. a. == deceive, ensnare. Alys. 258. K. Horn, 318
Bicharred, part. == deceived. Rel. Ant. ii. p. 211; M. Ode, 160. AS.
becýrran
Bicherme, v. a. == chirp about or around. O. and N. 279. AS. cyrm
Bick, v. n. == fight. Alys. 2337
Bicker, v. n. == quarrel. RG. 540. Fr. becquer. W. bicra == to fight
Bicker, sb. == a quarrel, contention, battle. RG. 538, 543
Biclipe, Biclupe, v. a. == accuse. 365 B.
—— == appeal. RG. 473
Biclose, v. a. == enclose. RG. 558, 218
Bid, v. a. == ask. RG. 77. 3 pl. pret. ‘badden.’ Alys. 5823. See ‘bede’
—— == command. RG. 29. pret. ‘bad.’ 683 B. part. ‘y-bede.’ RG.
383. AS. biddan](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-57-320.jpg)
![Bihede, v. a. == regard. O. and N. 635
Bihemmen, v. a. == cover, cloak. O. and N. 672
Bihepe, part. == heaped up. O. and N. 360
Bihete, v. a. == promise, pret. ‘bihet.’ RG. 381. ‘byheyghte.’ Alys.
3926
Bihoting, sb. == promise. Alys. 4000
Bike, sb. == cassia. Ps. xliv. 9. Literally ‘pitch.’ ON. bik
Bilace, part. == beset. Alys. 3357
Bilaue. See Bileve
Bilaucte. See Bilou
Bilede, v. a. == lead about. Pol. S. 155. O. and N. 68
Bilegge, v. a. == assert, allege. O. and N. 672
Bileve, v. a. == leave. RG. 421
—— v. n. == remain. RG. 372, 374. [bilaue]. Alys. 3541
Biliked, part. == rendered likely or probable. O. and N. 840
Bilime, v. a. == to mutilate. RG. 471, 560
Bilimp, v. n. == happen. M. Ode, st. 59 AS. belimpan
Bill, sb. (of a bird). O. and N. 79
—— == hatchet. Pol. S. 151
Bilou, pret. == laughed at. RG. 328. [bylaucte]. K. Horn, 681. [by
lowe], RG. 299. [by lowȝ]. RG. 64
Bimong, prep. == among. Wright’s L. P. p. 35
Bind, v. a. Wright’s L. P. p. 45. part. ‘ibounde.’ RG. 487](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-59-320.jpg)
![Binder, sb. HD. 2050
Binding, sb. == chain. Ps. cxxiv. 5
Bink, sb. See Bench
Bipahte, pret. == deceived. Rel. S. v. 128. AS. be-pæcan
Birade, v. a. == counsel. Alys. 3732
Birch, sb. == the tree. Alys. 5242
Bird, sb. RG. 177
Birde, sb. == lady. HD. 2760. A metathesis of ‘bride’
Birst, == bruised. Body and Soul, 86. AS. berstan.
Birth, sb. == nation. Ps. lxxviii. 10
Birthman, sb. == man of good birth. HD. 2101
Birthtime, sb. [burtyme]. RG. 9, 443
Birue, v. a. == rue, repent. Fragm. Sci. 325
Bis, sb. == purple. Wright’s L. P. p. 26. Fr. bis. Lat. byssus
Bisay, v. a. == recommend, say. RG. 422
Bisayen, == treated. See Besee.
Bischriche, v. a. == shriek at. O. and N. 67
Biscunien, v. a. == shun. M. Ode, 77
Bise, sb. == north wind. HD. 724. OHG. bísa
Bisend, v. a. == send after. RG. 491
Bishop, sb. RG. 376
Bishopric. RG. 414, 417](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-60-320.jpg)
![Bismere, sb. == blasphemy. Body and Soul, 110. [busemere]. RG.
12, 379. AS. bismér
Bisne, adj. == blind. O. and N. 78. AS. bisen
Bisoht, == sought out, got ready for. Pol. S. 220
Bisokne, sb. == beseeching. RG. 495
Bispel, sb. == proverb. O. and N. 127. AS. bispel
Bistad, sb. == a dwelling. Wright’s L. P. p. 38
Bistand, v. a. == stand by a person; hence, to press or urge them.
O. and N. 1436
Bistolen, part. == stolen, crept onwards. M. Ode, 9
Bisyhed, == the state of being busy. Alys. 3
Bit, sb. == a morsel. RG. 207
Bit, sb. == bottle. Ps. lxxvii. 13. [bite]. Body and Soul, 34. AS. bitte
Bitch, sb. Alys. 5394
Bite, v. a. Alys. 5435
Bite, sb. Alys. 5436
Bite, v. a. == drink. HD. 1731. Cf. bohem. ‘piti,’ potus; ‘pitka,’
potatio, &c. Gr. πίνω
Bitell, v. a. == excuse. O. and N. 263
Bitiȝt, == arrayed. O. and N. 1011. AS. biþæht. See Gloss. to Laȝ.
s. v.
Bitoȝe, == employed. O. and N. 702. AS. biteon. See Gloss. to Laȝ.
s. v.
Bitter, adj. Wright’s L. P. p. 87](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-61-320.jpg)
![Blaze, sb. 1254 HD. AS. blǽse, blýsan
Blear, v. n. == become bleareyed. Rel. Ant. ii. p. 211
Bleat, v. n. Ps. lxiv. 14
Bled, blete, sb. == foliage. O. and N. 1040, 57. AS. blæd
Bleed, v. n. RG. 560
Bleike, adj. == pale. HD. 470. AS. blác. ON. bleikr
Blench, sb. == a trick? O. and N. 378. ON. blekkja
Blench, v. n. == avoid (a thing). O. and N. 170
—— == flinch from [blinche]. 2184 B.
—— == deceive. Ritson’s AS. viii. 23
—— == give way? (of a ship) K. Horn, 1461. Another form of ‘flinch.’
AS. blinnan
Bleo, sb. == hue, complexion. O. and N. 152. Wright’s L. P. p. 35.
AS. bleo
Bless, v. a. RG. 406
Blessing, sb. RG. 421
Blete, adj. == bleak? O. and N. 616
Blete, sb. See Bled
Blike, v. n. == shine. Wright’s L. P. p. 52 AS. blícan
Blinch. See Blench
Blind, adj. RG. 376, 407
—— v. n. == become blind. Rel. Ant, ii. p. 211](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-63-320.jpg)
![Blink, sb. ‘to make blinks,’ == deride a person. HD. 307. See
Blench, sb.
Blin, v. n. == cease. RG. 566. pret. ‘blenyte.’ RG. 338. AS. blinnan
Bliss, sb. RG. 469
Blissful, adj. Wright’s L. P. p. 52
Blissfully, adv. Ps. xcvi. 1
Blithe, adj. RG. 15
Blithely, adv. 89 β
Blitheful. Ps. cxi. 5
Blive, adv. == quickly. RG. 544. See Belive
Blode, adj. == pale, dried up. Rel. Ant ii. p. 210. Germ. blöde
Blood, sb. RG. 388, 416
Bloody, adj. RG. 304, 311
Bloom, sb. HD 63
Bloom, v. n. Ps. xxvii. 7
Blote, adj. == dried. Rel. Ant. ii. p 176
Bloute, v. n. == swell out? HD. 1910. ON. blautr. Eng. bloat
Blow, v. a. == as ‘blow the fire.’ HD. 385. Alys. 5030
—— v. n. pret. ‘blew.’ 524 β
Blow, vb. n. part. ‘blowe,’ == blown, in blossom. O. and N. 1634
Blowing, sb. 467 β
Blue, adj. [blo]. Wright’s L. P. p. 86](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-64-320.jpg)
![Bo, == be. O. and N. 166, et passim. See Be
Bo, == both. q. v.
Boar, sb. RG. 133
Board, sb. == table. 122 β.; plank. Alys. 6415
Boast, sb. RG. 258. pomp. St Swithin, 43
Boast, v. n. Alys. 2597
Boasty, adj. == boastful. Fragm. Sci. 283
Bobance, sb. == boasting. Pol. S. 189 Fr. bobance
Bochevampe, (sic in MS.). == botched vamps or fronts of shoes.
Rel. Ant. ii. p. 176
Bode, sb. == commandment. Ps. cxviii. 134, 128, et passim
Bode, v. a. == foretell. O. and N. 530
Boded ? Pol. S. 152
Boding, sb. RG. 416, 428
Bodeword, sb. == message. Ps. ii. 6
Body, sb. RG. 395, 547
Boffing, == swelling or puffing. RG. 414. Fr. buffer, to puff the
cheeks
Boistous, adj. == coarse, rude. Alys. 5660. [boustes] Fragm. Sci.
273
Bold, sb. == a building. RG. 44. AS. bold
Bold, v. a. == embolden. Alys. 2468. [bald]. Ritson’s AS. viii. 128
—— adj. RG. 383. ‘bolder.’ RG. 465](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-65-320.jpg)
![Boldhede, == boldness. O. and N. 514
Boldly, adv. RG. 500, 19
Boleax, sb. == large axe. Rel. Ant. ii. p. 176. ON. bolöxi
Bolken, v. n. == belch. Ps. cxliii. 13
Bollen, == swollen. Body and Soul, 31. ‘ibolȝe.’ O. and N. 145
Bolster, sb. Rel. S. v. 90
Bolt, sb. ‘ȝoure bolt is sone ischote.’ St Kath. 54
Bonde, sb. == bondman. Pol. S. 150
Bondman. RG. 370. HD. 32
Bone, sb. == os. RG. 446
Bone, sb. == prayer. RG. 14. AS. bén. SS. bone
Boned, [y-boned] == having bones. RG. 414
Bonére, adj. == debonair, graceful. Alys. 6732
Bonny, adj. Alys. 3903
Book, sb. RG. 374, 420
Boot, sb. == use, avail. Body and Soul, 92
—— == remedy, means (bote). RG. 277, 408. Pilate, 139
Booth, sb. Alys. 3457
Booze, sb. [bous] == drink. Wright’s L. P. p. 111. Dutch, buysen
Booze, v. n. == drink. Rel. Ant. ii. p. 175
Bord, sb. == border. Alys. 1270
Borough, sb. [boru]. RG. 72](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-66-320.jpg)
![Borow, v. a. == defend. Wright’s L. P. pp. 24, 25. part. ‘iborȝe,’ O.
and N. 881
Borow, sb. == surety. RG. 472, 497
Borrow, v. a. RG. 393
Borstax, sb. == pick-axe. Pol. S. 151
Bosk, sb. == wood. RG. 547. Fr. bos, bosche
Boss, sb. == an ornament of dress. Pol. S. 154. Fr. bosse
Bote, sb. See Boot
Botemay, sb. == bitumen. Alys. 4763
Botfork, sb. == a crooked stick. Wright’s L. P. p. 110
Both, adj. RG. 376, 445. ‘both two.’ Body and Soul, 120. [bo].
Wright’s L. P. p. 58
Both, == are. See Be
In O. and N. 630, 633, the meaning of ‘both’ is uncertain; perhaps a
mistake for ‘doth’
Botheler, sb. == peasant, shepherd. Body and Soul, 144; from
‘booth’?
Boting, sb. == recompense. Alys. 5711
Bough, sb. [bowe]. RG. 283. [boye], O. and N. 15
Bouk, sb. == body. Alys. 3946. [buc], O. and N. 1130. AS. búce.
Germ. bauch
Bouked, adj. == protuberant. Alys. 6265
Boulder, sb. == a large stone. HD. 1790](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-67-320.jpg)
![are found in the O. Norse, while the AS. ‘bredan’ apparently is only
used in those marked with an asterisk.
* I. act. to weave, part. ‘broiden.’ O. and N. 645
II. act. to move a thing from its place. Hence,
α. to draw out, as a sword. HD. 1825. part. ‘ybrad’ == drawn,
caught. Wright’s L. P. p. 39
β. to brandish, as a sword or spear. Alys. 7373
γ. to pull down. RG. 22. [breide], Alys. 5856
* δ. to seize, or perhaps tear. Rel. S. v. 200. [brede]
III. neut. to change, as—
α. to awake out of sleep. HD. 1282
β. of any violent motion of body, as to leap. Body and Soul, 46
Braid, sb. ==
1. a quick motion, from III. β.; hence, ‘at a breid’ == in an instant.
Body and Soul, 182. ON. bragð.
2. a violent struggle or wrench. RG. 22
Brain, sb. RG. 49, 446
Branch, sb. RG. 152
Brand, sb. == a burning mass. Body and Soul, 208
—— == torch. Alys. 5295. [brond]. AS. brand
—— == fire. Alys. 1856. [wilde bround]
Brased, adj. == of brass. Ps. cvi. 16
Brass, sb. RG. 2, 251](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-69-320.jpg)
![Bray, sb. == noise. Alys. 2175
Breach, sb. [bruch]. Wright’s L. P. p. 30
Bread, sb. RG. 238
Breadth, sb. [brede]. RG. 385
Break, v. a. 47 B. part. ‘i-broke’ 1005 B.
—— v. n. pret. ‘brake.’ 2154 B.
—— == to break out (of flesh). 2421 B.
Breaking, sb. == breach, gap. Ps. cv. 23
Breast, sb. RG. 419
Breath, sb. Fragm. Sci. 203
Breathe, v. n. Fragm. Sci. 202
Breche, == beech? q. v.
Breech, sb. == rump. RG. 322
—— == breeches. 260 B.
Breed, v. a. (of a bird). 2 s. pres. ‘breist.’ O. and N. 1631. RG. 177.
part. ‘ibred’ == brought up, educated. O. and N. 1722. Body and
Soul, 81
Breed, v. n. == spring forth. Wright’s L. P. p. 45
Breist, == breedest. See Breed
Breme, adj. == glorious, renowned. Wright’s L. P. pp. 52, 32. AS.
breme
—— == eager, lustful. O. and N. 202
Brenne, sb. == burning. HD. 1239](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-70-320.jpg)
![Breth, sb. == wrath. Ps. ii. 5; vi. 2. ON. brædi == anger
Breven, v. a. == write down. Pol. S. 156.
Brew, v. a. [browe]. RG. 26
Brewer, sb. Rel. S. vii. 35
Brewster, sb. Rel. Ant. ii. p. 176
Breȝe, sb. See Brow.
Breze, sb. == gadfly. Ps. civ. 34. AS. brimse
Briar, sb. RG. 331
Bridal, sb. Alys. 1071. K. Horn, 1064
Bride, sb. HD. 2131
Bridegroom, sb. [bridegome]. Ps. xviii. 6
Bride, sb. == bridle. Alys. 7626
Bridge, sb. RG. 399
Bridle, sb. RG. 396
Bright, adj. HD. 2131. Wright’s L. P. p. 33
Brim, sb. == brink. 476 β
Brimstone, sb. Body and Soul, 219
Bring, v. a. RG. 379. pret. ‘brought.’ RG. 309. part. ‘ybroȝt,’ ‘ibrouȝt.’
RG. 376, 491
Brinie, sb. == cuirass. HD. 1775. Fr. brugne, brugnie. The root is
‘brun’ from ‘brinnan,’ to burn or shine; Cf. OHG. brunna
Brink, sb. Alys. 3491. K. Horn, 147
Brise, v. a. == bruise. HD. 1835](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-71-320.jpg)
![Bristle, sb. Alys. 6621
Bristled, adj. == having bristles. Alys. 5722
Britheling, == worthless, a rascal. Rel. S. vi. 11. Cf. O. Eng.
‘brothell’
Brittene, == cut in pieces? HD. 2700. Cf. ‘brittned,’ in Gloss. to
Ormulum. AS. bryttian
Broach, sb. (an ornament). RG. 489. Alys. 6842
Broad, adj. RG. 1. [brede], O. and N. 963
—— v. a. == make broad, part. ‘ibroded.’ O. and N. 1310
Broerh, adj. == brittle? Wright’s L. P. p. 23
Brood, sb. RG. 70
Broodful, adj. Ps. cxliii. 13
Brook, sb. RG. 80
Broom, sb. (genista). Alys. 2492
Brost, sb. O. and N. 976, a mistake for ‘prost,’ i.e. ‘priest.’ The Jesus
Coll. MS. reads ‘preost’
Broth, sb. RG. 528
Brother, sb. RG. 371, 478
Brouke, v. a. == use, enjoy. HD. 311 AS. brúcan. Germ. brauchen
Brow, sb. Wright’s L. P. p. 28. [breȝe]. Ib. p. 34
Brown, adj. RG. 429
—— v. n. == become brown. Alys. 3293
Brun, sb. == a brown jar. K. Horn, 1134](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-72-320.jpg)
![Bullock, sb. Ritson’s AS. iii. 8
Bunting, sb. (the bird). Wright’s L. P. p. 40
Burde, sb. == beard. Alys. 1164
Burdon, sb. == a pilgrim’s staff. K. Horn, 1093. Fr. bourdon
Burel, sb. == sackcloth. Alys. 5475. Pol. S. 221. Fr. bure, burel. See
Roq.
Burgess, sb. RG. 540, 541
Burial, sb. See Buryel
Burn, v. a. pret. ‘barnde.’ RG. 380, 511. ‘brende.’ RG. 536. part.
pres. ‘berninde.’ RG. 534
Burn, sb. == rivulet. O. and N. 916. AS. byrnan, to burn. Cf. Lat.
torrens, from torreo
Burst, v. n. pret. ‘barst.’ RG. 437
Burst, sb. == injury. Wright’s L. P. p. 24. AS. byrst
Burthen, sb. HD. 807
Bury, v. a. RG. 123. part. ‘y-bured.’ RG. 382. AS. byrgan
Burying, sb. RG. 382. [beryng]. Alys. 4624
Buryels, sb. == a tomb, grave. RG. 204. AS. byrgels
Busemere, == blasphemy. See Bismere
Busily, adv. Ps. cxlii. 7
Busk, v. a. == array. Pol. S. 239
Busy, adj. Alys. 3906
But, adv. 43 B.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-74-320.jpg)
![But, prep. == except [bote]. RG. 382. [butent]. Rel. S. ii. 25
—— == without [bute]. O. and N. 184. AS. bútan
But, sb. == a put, i.e. cast or throw, HD. 1040
But, part. == contended. HD. 1916. Fr. bouter
Butcher, sb. Pol. S. 192
Bute, prep. See But
Butler, sb. RG. 187, 438
Butte, sb. == a fish, probably a turbot. HD. 759. The Prompt. Parv.
translates it by ‘pecten;’ the Pictorial Vocab., published by Mr Wright,
p. 254, has ‘hic turbo’ == ‘a but.’ See N. and Q. 2d S. vi. 382. Sw.
butta
Butter, sb. HD. 643
Button, sb. Pol. S. 239
—— v. n. == break out. St Swithin, 151. Fr. boutonner. Cotgr.
Buxomness, sb. == obedience. RG. 234, 318. AS. buhsomnes,
from ‘bugan,’ to bow
Buy, v. a. [biggen]. Moral Ode, st. 33. [buggen]. O. and N. 1366.
pret. ‘bouȝte.’ RG. 379, 496. ‘bu,’ imper. RG. 390
—— == to exact atonement for. K. Horn, 912
—— == redeem. Ps. xxv. 11
Buyer, sb. == redeemer. Ps. xviii. 15
Buzzard, sb. Alys. 3049
By, prep. == beside (of place). 1213 B. ‘Nolde God that ich bi thé
sete’](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-75-320.jpg)
![C.
Cable, sb. RG. 148
Cacherel, sb. == catch poll. Pol. S. 151
Cage, sb. Alys. 5011
Caitiff, sb. Body and Soul, 229
Cake, sb. Cok. 55
Cales, sb. == a kind of serpent. Alys. 7094
Calf, sb. (the animal). Alys. 6351
Call, v. n. Wright’s L. P. p. 59
Call, sb. == cap worn on the head. Pol. S. 158. Fr. cale
Caluȝ, adj. == bald. Alys. 5950. AS. calo, caluw
Camel, sb. Alys. 854
Can, vb. == am able [con]. Wright’s L. P. p. 82. [cunne]. 2 s. pres.
‘cost.’ Wright’s L. P. p. 91. O. and N. 47. pret. ‘cowþe.’ RG. 29
—— == know [con]. RG. 443. [cunne]. O. and N. 48. 2 s. pres.
‘canst.’ O. and N. 560
Candle, sb. RG. 290, 561
Candlemas, sb. St Dunstan, 3
Canel, sb. == cinnamon. Wright’s L. P. p. 27. Fr canelle. Lat. canna
Cankerfret, adj. RG. 299
Canon, sb. RG. 510](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-77-320.jpg)
![Chamberlain. RG. 390, 490
Champion, sb. HD. 1015
Chance, sb. == condition, fortune. RG. 465
—— == chance [cheance]. RG. 210
Chancellor, sb. RG. 540, 468
Chancellory, sb. == office of chancellor. 452 B.
Chane, vb. pret. == cleft. Alys. 2228. AS. cínan. perf. cán. The ‘ch’
appears in ‘tochan,’ the pret. of ‘tocínan,’ in Laȝamon, ii. 468. Weber
wrongly derives the word from Fr. choir, and makes it mean ‘fell’
Change, sb. RG. 493
Change, vb. a. RG. 548
Chantment, sb. == enchantment. RG. 28, 149
Chapel, sb. RG. 472, 473
Chapitle, sb. == chapter of a cathedral RG. 473
Chaplain, sb. 961 B.
Chapman, sb. RG. 539
Chapter, sb. (of a cathedral). 601 B.
Char, sb. == turn, movement. Body and Soul, 79. Hence ‘ȝeynchar’
== repentance. Wright’s L. P. p. 46. SS. charren. AS. cérran, cérre.
Germ. kehren
Charge, v. a. == load. RG. 13. part. ‘icharged.’ Pol. S. 195
—— sb. == load, weight. RG. 416
—— == expense. RG. 189
Charity, sb. Pol. S. 202. ‘par charité.’ 1811 B.](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-81-320.jpg)
![Charm, sb. == spell. Alys. 81
Charming, sb. == spell. Alys. 404
Charreye, sb. == car. Alys. 5097
Charter, sb. RG. 477, 498
Chase, sb. == hunting. RG. 6
Chaste, adj. 154 B.; [cheste]. Alys. 7050. ‘chaster.’ RG. 191
Chaste, v. a. == chastise. RG. 134
Chastise, v. a. RG. 420
Chasuble, sb. == priest’s robe. 953 B. Fr. casule. Ital. casupola
Chasur, sb. == horse for hunting. Signa ante Jud. 110. Fr. chaceor
Chaterestre, sb. == a female chatterer. O. and N. 655
Chattels, sb. [chateus]. RG. 471, 569. Another form of ‘cattle’
Chattering, sb. O. and N. 744
Chaumpebataile, sb. == battle-field. Alys. 5553
Chavling, sb. == jawing. O. and N. 284, 296
Chawl, sb. == jaw. Body and Soul, 189. Pol. S. 154. AS. ceafl. SS.
chevele. pl. chæfles
Chawl, v. n. == to chide, jaw. Pol. S. 240
Cheap, sb. == haggling? Wright’s L. P. p. 39
Cheap, v. a. == buy. Pol. S. 159. AS. ceápian
Cheaping, sb. == market. Pol. S. 151
Cheek, sb. Wright’s L. P. p. 34](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-82-320.jpg)
![—— == principal. St Swithin, 22
Chieftain, sb. [cheventeyn]. RG. 386, 400
Chilce, sb. == childishness? M. Ode, 4. Formed from ‘child,’ as
‘milce’ from ‘mild’
Child, sb. RG. 392, 441; [chil]. O. and N. 1438
Child, v. n. == bring forth a child. Alys. 604
Childbed, sb. RG. 379
Childering, sb. == bringing forth a child. Rel. S. ii. 7
Chill, sb. RG. 7
Chimbe, sb. == cymbal. Ps. cl. 5
Chime, sb. (of bells). Alys. 1852. Dan. kime
Chin, sb. 522 β
Chinche, adj. == niggardly. HD. 1763. Fr. chice == avarice
Chirchegong, == churchgoing. RG. 380. Cf. ‘idelgong’
Chirm, sb. == chirping and screaming of birds. O. and N. 305. AS.
cyrm
Chirurgeon, sb. RG. 566
Chivalry, sb. == prowess. RG. 413
Chivauché, sb. == an expedition, a body of men. Ritson’s AS. viii.
141. Fr. chevauchée, from cheval.
Choice, sb. RG. 111
Chokering, sb. == a low chattering. O. and N. 504
Cholle, == shall. RG. 379, in the compound form ‘ycholle’](https://image.slidesharecdn.com/1130775-250520204500-d0349def/85/Normal-Approximation-And-Asymptotic-Expansions-Clasics-In-Applied-Mathmatics-Rabi-N-Bhattacharya-84-320.jpg)
Normal Approximation And Asymptotic Expansions Clasics In Applied Mathmatics 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.
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- 10. Normal Approximation and Asymptotic Expansions ci Rabi N. Bhattacharya University of Arizona Tucson, Arizona R. Ranga Rao University of Illinois at Urbana-Champaign Urbana, Illinois Society for Industrial and Applied Mathematics Philadelphia
- 11. Copyright © 2010 by the Society for Industrial and Applied Mathematics This SIAM edition is an updated republication of the work first published by Robert E. Krieger Publishing Company, Inc., in 1986, which was an updated and corrected version of the original edition that was published by Wiley in 1976. 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- Normal approximation and asymptotic expansions / Rabi N. Bhattacharya, R. Ranga Rao. p. cm. -- (Classics in applied mathematics ; 64) "Updated republication of the work first published by Robert E. Krieger Publishing Company, Inc., in 1986"--Copr. p. Includes bibliographical references and index. ISBN 978-0-898718-97-3 (pbk.) 1. Central limit theorem. 2. Convergence. 3. Asymptotic expansions. I. Ranga Rao, R. (Ramaswamy), 1935- II. Title. QA273.67.B48 2010 519.2--dc22 2010031917 S.LaJTL is a registered trademark.
- 12. To owri and hantha
- 14. Contents PREFACE TO THE CLASSICS EDITION xiii PREFACE xv LIST OF SYMBOLS xvii CHAPTER 1. WEAK CONVERGENCE OF PROBABILITY MEASURES AND UNIFORMITY CLASSES I 1. Weak Convergence, 2 2. Uniformity Classes, 6 3. Inequalities for Integrals over Convex Shells, 23 Notes, 38 CHAPTER 2. FOURIER TRANSFORMS AND EXPANSIONS OF CHARACTERISTIC FUNCTIONS 39 4. The Fourier Transform, 39 5. The Fourier—Stieltjes Transform, 42 6. Moments, Cumulants, and Normal Distribution, 44 7. The Polynomials Ps and the Signed Measures Ps , 51 8. Approximation of Characteristic Functions of Normalized Sums of Independent Random Vectors, 57 9. Asymptotic Expansions of Derivatives of Characteristic Functions, 68 10. A Class of Kernels, 83 Notes, 88 ix
- 15. X Contents CHAPTER 3. BOUNDS FOR ERRORS OF NORMAL APPROXIMATION 90 11. Smoothing Inequalities, 92 12. Berry—Esseen Theorem, 99 13. Rates of Convergence Assuming Finite Fourth Moments, 110 14. Truncation, 120 15. Main Theorems, 143 16. Normalization, 160 17. Some Applications, 164 18. Rates of Convergence under Finiteness of Second Moments, 180 Notes, 185 CHAPTER 4. ASYMPTOTIC EXPANSIONS- NONLATTICE DISTRIBUTIONS 188 19. Local Limit Theorems and Asymptotic Expansions for Densities, 189 20. Asymptotic Expansions under Cramer's Condition, 207 Notes, 221 CHAPTER 5. ASYMPTOTIC EXPANSIONS—LATTICE DISTRIBUTIONS 223 21. Lattice Distributions, 223 22. Local Expansions, 230 23. Asymptotic Expansions of Distribution Functions, 237 Notes, 241 CHAPTER 6. TWO RECENT IMPROVEMENTS 243 24. Another Smoothing Inequality, 243 25. Asymptotic Expansions of Smooth Functions of Normalized Sums, 255
- 16. Contents xi CHAPTER 7. AN APPLICATION OF STEIN'S METHOD 260 26. An Exposition of Gotze's Estimation of the Rate of Convergence in the Multivariate Central Limit Theorem, 260 APPENDIX A.I. RANDOM VECTORS AND INDEPENDENCE 285 APPENDIX A.2. FUNCTIONS OF BOUNDED VARIATION AND DISTRIBUTION FUNCTIONS 286 APPENDIX A.3. ABSOLUTELY CONTINUOUS. SINGULAR, AND DISCRETE PROBABILITY MEASURES 294 APPENDIX A.4. THE EULER—MACLAURIN SUM FORMULA FOR FUNCTIONS OF SEVERAL VARIABLES 296 REFERENCES 309 INDEX 315
- 18. Preface to the Classics Edition It is with great pleasure that the authors welcome the publication by SIAM of the edited reprint of Normal Approximation and Asymptotic Expansions. The original edition was published in 1976 by Wiley, followed by a Russian transla- tion in 1982 and an edited version with a new chapter by Krieger in 1986. The book has been out of print for nearly twenty years. Statistical applications such as "higher order" comparisons of efficiency, and the evaluation of the improvement over the classical central limit theorem due to the widely popular and important bootstrap methodology of Efron, have led to a renewed interest in the subject matter of the book. We also note with a measure of happiness that the theory of asymptotic expansions for sums of weakly dependent random variables/vectors due to Gotze and Hipp made use of some of the formalism and estimation in our book. We have controlled an initial impulse to present this theory, as it would make the book substantially increase in size and take us much time to ready it for publication. Keeping to independence, however, a short new chapter is added on Gotze's application to the multivariate CLT of an ingenious method of Stein. The exposition, and a somewhat modified treatment presented here of the rather difficult original paper, resulted from a collaboration between one of the authors and Professor Susan Holmes. Finally, we are deeply appreciative that our colleague Professor William Faris has always liked the book. His support, as well as that of SIAM acquisitions editor Sara Murphy, made the publication of the present reprint possible. We are indebted to Bill and Sara. xiii
- 20. Preface This monograph presents in a unified way various refinements of the classical central limit theorem for independent random vectors and in- cludes recent research on the subject. Most of the multidimensional results in this area are fairly recent, and significant advances over the last 15 years have led to a fresh outlook. The increasing demands of application (e.g., to the large sample theory of statistics) indicate that the present generality is useful. It is rather fortunate that in our context precision and generality go hand in hand. Apart from some material that most students in probability and statistics encounter during the first year of their graduate studies, this book is essentially self-contained. It is unavoidable that lengthy computations frequently appear in the text. We hope that in addition to making it easier for someone to check the veracity of a particular result of interest, the detailed computations will also be helpful in estimations of constants that appear in various error bounds in the text. To facilitate comprehension each chapter begins with a brief indication of the nature of the problem treated and its solution. Notes at the end of each chapter provide some history and references and, occasionally, additional facts. There is also an Appendix devoted partly to some elementary notions in probability and partly to some auxiliary results used in the book. We have not discussed many topics closely related to the subject matter (not to mention applications). Some of these topics are "large deviation," extension of the results of this monograph to the dependence case, and rates of convergence for the invariance principle. It would take another book of comparable size to cover these topics adequately. We take this opportunity to thank Professors Raghu Raj Bahadur and Patrick Billingsley for encouraging us to write this book and giving us xv
- 21. xvi Preface advice. We owe a special debt of gratitude to Professor Billingsley for his many critical remarks, suggestions, and other help. We thank Professor John L. Denny for graciously reviewing the manuscript and pointing out a number of errors. We gratefully acknowledge partial support from the National Science Foundation (Grant. No. MPS 75-07549). Miss Kanda Kunze and Mrs. Sarah Oordt, who did an excellent job of typing the manuscript, have our sincere appreciation. R. N. BHATTACHARYA R. RANGA RAO Note In this reprinted edition a new chapter (Chapter 6) has been added and misprints in the original edition corrected.
- 22. List of Symbols AB set of all elements of A not in B: (1.4) A+y (x+y:xEA): (5.5) A ` set of all points at distances less than e from A : (1.17) A -` set of all x such that the open ball of radius a centered at x is contained in A: (2.38) 9 a generic class of Borel sets c a* (d : µ), special classes of Borel Sets: d. (d : 41o, v) (17.3), (17.52) a„ (14.64) a, /3 usually nonnegative vectors with integral coordinates; sometimes positive numbers ^aj sum of coordinates of a nonnegative integral vector B a generic Borel set B, B„ positive square roots of the inverses of matrices V, V,,: (9.7), (19.28) B (x : e) open ball of radius a centered at x: (1.10) n1'` Borel sigma-field of R k Cl(A) closure of A C class of all convex Borel subsets of R k c(B) convex hull of B: Section 3 xvti
- 23. xviii List of Symbols cov(X, Y) covariance between random variables X, Y: (A.1.5) Cov(X) matrix of covariances between co- ordinates of a random vector X: Appendix A. 1 D average covariance matrix of centered truncated random vectors Z,,...,Z,,: (1-4.5) D° ath derivative: (4.3) d (0, aA) euclidean distance between the origin and aA: Section 17 d0(G1 , G2) (17.50) d p Prokhorov distance: (1.16) dBL bounded Lipschitzian distance: (2.51) d(P,4) (12.47) Det V, Det D determinant of a matrix V or D det L absolute value of the determinant of the matrix of basis vectors of a lattice L: (21.20) 0(A, B) Hausdorff distance between sets A and B: (2.62) (14.4) 0,,,(e) (14.105), (14.106) Or s (15.7) ^^ J (17.55) S,!s (18.4) aA topological boundary of A: (1.15) EX, E(X) expectation or mean of a random variable or random vector X: (A. 1.2), (A. 1.3) E, E generic small numbers E symbol for "belongs to" f Fourier transform of a function f: (4.5) f (4.4) f(x) f(x+y): (11.5) f*g convolution of functions f and g: (4.9) f*" n-fold convolution of a function f: (4.11)
- 24. List of Symbols xix a generic class of Borel-measurable functions fundamental domain for the dual lattice L*: (21.22) normal distribution on R k with zero mean and identity covariance matrix density of 1 (Dm. v normal distribution with mean m and covariance V m V density of 4,,, v : (6.31) (D,o (15.5), (18.10) G., m , g0 , a special probability measure and its density: (10.7) gT (16.7) Y(f:E), Y *(f:E) (11.8), (11.18) 11i ^ (9.8), (19.32) I k X k identity matrix IA indicator function of the set A Int(A) interior of A K( a smooth kernel probability measure assigning either all or more than half its mass to the sphere B (x : e): (11.6), (11.16), (15.26) X„ with cumulant, average of with cumulants of X,,...,X,,: (6.9), (9.6), (14.1) average of with cumulants of centered truncated random vectors Z,, ... , Z,,: (14.3) X,,,j , X,,.,, with cumulant of X^, I < j < n, and their average: (9.6), Sections 19, 20 X,(z) (6.16) L a lattice: Section 21 L* lattice of periods of f, f being the char- acteristic function of a lattice random vector: (21.9), (21.19) L(c,d) a Lipschitzian class of functions: (2.50) 11,11 Liapounov coefficient: (8.10) A, A smallest and largest eigenvalues of an average covariance matrix V: Section 16
- 25. xx List of Symbols Ak Lebesgue measure on R k A,.,,(F) (23.8) M,(f), MO(f) (15.4) M1(x : e), mj(x : E) supremum and infimum of f in B (x : e): (11.2) )i, set of all finite signed measures on a metric space. µ+, µ-, ^ µI positive, negative, and total variations of a finite signed measure µ: (1.1) Il µdl variation norm of a signed measure µ: (1.5) µ Fourier-Stieltjes transform of µ: (5.2) µ• v convolution of two finite signed measures µ, v: (5.4) µ•" n-fold convolution of µ: (5.6) µo T -' signed measure induced by the map T: (5.7) µQ ath moment, average of ath moments of X1 ,...,X,,: (6.1), (14.1) average of ath moments of centered truncated random vectors Z,,...,Z,,: (14.3) µ.(t), f3,(t) (8.4) v! PI!v2!...Pk! where v=(P,,••.,Pk) is a non- negative integral vector v,, vo special signed measures: (15.5) P a probability measure, a polyhedron set of all probability measures on a metric space P characteristic function of a probability measure P: (5.2) P,(z : (x,)) a special polynomial in z: (7.3) P.(-40, v : {x,}) a polynomial multiple of 0o.v : (7.11) P,( - 'o, v : {x,}) signed measure whose density is P,( - $o,v: (X„}) Pa a special polyhedron: (3.19) point masses of normalized lattice random vectors X 1 ,.. .,X17 : (22.3) P;,(Y.,,,) point masses of normalized truncated lat- tice random vectors: (22.3)
- 26. List of Symbols xxi Q„ distribution of n - '/Z(XI + • • • where X 1 ,...,X,, are independent random vectors having zero means and average covariance matrix V (or 1) Q„ distribution of n - 1 2(Y, + + Y„), where YT's are truncations of XD's: (14.2) Q„ distribution of n -1 '2(Z1 + + Z„), where Z^=Y^—EYE : (14.2) 9n,m , 9n,m local expansions of point masses of Q, Q,,' in the lattice case: (22.3), (22.38), (23.2) p(x, A) distance between a point x and a set A: (1.18) p, sth absolute moment, average of sth absolute moments of X 1 ,.. .,X: (6.2), (9.6), (14.1) p; average of sth absolute moments of centered truncated tandom vectors (14.3) p absolute moment of X^, I < j < n, and their average: (14.1), (17.55) SS, S. special periodic functions: (A.4.2), (A.4.14) S Schwartz space: (A.4.13) ak _ I surface area measure on the unit sphere of R k: Section 3 11 T h norm of a matrix T: (14.17) Tr (16.6) T(f : 2E), T*(f : 2E) (11.8), (11.18) V average of covariance matrices of ran- dom vectors X 1 ,...,X: (9.6), (14.5) wf(A) oscillation off on A: (2.7), (11.1) wf(x : E) oscillation of f on B (x : E): (2.7), (11.3) Zj(E : µ) average modulus of oscillation off with respect to a measure µ: (11.23) sup 1 (: s): (11.24) y IxI x,I+ • • • +Ixk I, where x=(x l ,...,xk): (4.8)
- 27. xxii List of Symbols ya,n+ ya,n (22.3) z+ set of all nonnegative integers (Z + )k set of all k-tuples of nonnegative integers il•Ii, <,> euclidean norm and inner product II•IIP Lp-norm z set of all integers
- 28. CHAPTER 1 Weak Convergence of Probability Measures and Uniformity Classes Let Q be a probability measure on a separable metric space S every open ball of which is connected (e.g., S=R"). In the present chapter we characterize classes f of bounded Borel-measurable functions such that sup If fdQ„- f fdQI-0 (n--*oo), (1) Jeff S S for every sequence { Q„ : n> I) of probability measures converging weakly to Q. Such a class is called a Q-uniformity class. It turns out that is a Q uniformity class if and only if sup w1 (S)<oc, urn fsup f wj (x:e)Q(dx)l =0, (2) JE c 4° LJE T-, S J where wJ(S) is the (total) oscillation off on S. and w1(x : e) its oscillation on the open ball of radius a centered at x. This suggests that the ap- propriate characteristics of f on which the rate of convergence ffdQ„ -. ffdQ depends are (i) w1(S) and (ii) the average oscillation function €.- fwJ(x : e)Q (dx). Specialized to indicator functions of Borel sets A, this says that the rate of convergence Q„(A)-► Q(A) depends on the function E_*Q((8A)`), where aA is the boundary of A and (8A) is the set of all points whose distances from aA are less than E. We have pursued this line of thinking in Chapters 3 and 4 to obtain appropriate rates of convergence for the central limit theorem. Section 1 contains a brief review of those aspects of weak conver- gence theory that are relevant for proving results on characterization of
- 29. Weak Convergence and Uniformity Classes uniformity classes in Section 2. These two sections are not used in the sequel (except as motivation). In Section 3 we obtain estimates such as sup 4((aC))<d(k)e (e>0), (3) CEC where 4) is the standard normal distribution on R', C is the class of all (Borel-measurable) convex subsets of R k, and d(k) is a positive number depending only on k. We have several occasions in Chapters 3 and 4 to use these estimates for deriving rates of convergence Q„(C)-->4'(C), C E C', where Q„ is the distribution of the normalized sum of n independent random vectors. 1. WEAK CONVERGENCE In this section we briefly review some standard results in the theory of weak convergence of probability measures. Throughout this section S denotes a metric space with a metric p. The Borel sigma field 3 of S is the smallest sigma-field containing the class of all open subsets of S. We say is a (signed) measure on S if it is a (signed) measure defined on . The class of all finite signed measures on S is denoted by OiL, and the subclass of O1i, comprising all probability mea- sures is denoted by 'P. Given a finite signed measure s on S, one defines three associated set functions called the positive, negative, and total variations of µ, respectively, by (B)=sup {µ(A):ACB, AE), (B)= µ - (B)=— inf( (A):ACB, AE°.1J), (BEAU) (1.1) 1µ1=µ++ tL . The so-called Jordan-Hahn decompositions asserts that µ+ and s - (and, therefore I s) are finite measures on S satisfying (1.2) For every finite signed measure s on a separable metric space S, we define the support of as the smallest closed subset of S whose complement has I µI-measure zero: that is, support of µ= n {F:F closed, IµI(S F)=0}, (1.3) tSee Halmos [1], pp. 121-123.
- 30. Weak Convergence 3 where for any two sets A, B we write AB={x:xEA,x^B}. (1.4) Note that the separability of the metric space S ensures that the comple- ment of the right side of (1.3) has zero I SI-measure. The class OIL of (set) functions on J into R 1 is a real linear space with respect to pointwise addition and multiplication by real scalars. It is a Banach space when endowed with the variation norm IIµII=IIl(S) (ttEOIL). (1.5) Let C(S) denote the class of all complex-valued, bounded, continuous functions on S. The weak totopogy on Olt, is the weakest topology (on OIL) that makes the maps µ— f fdµ [f EC(S)] (1.6) on cX into the complex field C continuous. The right side of (1.6) always stands for the Lebesgue integral of (a µ-integrable, complex-valued, Borel- measurable function) f on S. The Lebesgue integral of f on a Bore! set B is denoted by fBfdµ. (1.7) When it becomes necessary to indicate the variable of integration, we also write f f (x) µ(dx) (1.8) instead of f f dµ. In this monograph we are particularly concerned with the relativized weak topology on the class 6P of all probability measures on S. In this topology convergence of a sequence (Q„) of probability measures to a probability measure Q means lira f fdQ„= f fdQ (1.9) for every f in C(S). The following theorem gives several characterizations of weak convergence of a sequence of probability measures.
- 31. 4 Weak Convergence and Uniformity Classes THEOREM 1.1. Let S be a metric space. Let Q. (n= 1,2,... ), Q be probability measures on S. The following statements are equivalent. (i) Q„ converges weakly to Q. (ii) lin en f f dQ,, = ffdQ dQ for every uniformly continuous f in C (S). (iii) urn Q. (F) < Q (F) for every closed subset F of S. (iv) lint Q„(G)> Q(G) for every open subset G of S. n For a proof of this theorem we refer to Billingsley [1] (Theorem 2.1, pp. 11-14) or Parthasarathy [1] (Theorem 6.1, pp. 40-42). Let B (x : E) denote the open ball with center x and radius E, B(x:E)={y:yES, p(x,y)<E} (xES, c>0). (1.10) For an arbitrary real-valued function f on S we define, for each positive number e, the oscillation function wj(- wf(x:E)= sup {I f(z)-f(y)I:y,zEB(x:c)} (x ES). (1.11) For a complex-valued function f= g + ih (g, h real-valued), define Wf(X:E)— wg (X:E)+Wh(X:E) (xES, E >0). (1.12) The oscillation function is Borel-measurable on the (Borel-measurable) set on which it is finite.t The set of points of discontinuity of f is Borel- measurable and can be expressed as (x:wj (x: 1 ) +0 as n-moo}. (1.13) n 1 Now let Q be a probability measure on S. A complex-valued function f on S is said to be Q-continuous if its points of discontinuity comprise a set of Q-measure zero. In particular, if the indicator function IA of a set A, taking values one on A and zero on the complement of A, is Q-continuous, we say A is a Q-continuity set. Since the set of points of discontinuity of IA is precisely the boundary 8A of A, A is a Q-continuity set if and only if Q(aA)=0. (1.14) Recall that the (topological) boundary aA of a set A is defined by aA =Cl(A)Int(A), (1.15) tSee relations (11.1)—{11.4) and the discussion following them.
- 32. Weak Convergence 5 where C1(A), Int(A) are the closure and interior of A, respectively. LEMMA 1.2. Let Q be a probability measure and f a complex-valued, bounded, Borel-measurable function on a metric space S. The following statements are equivalent. (i) f is Q-continuous. (ii) lim Q((x:wf(x:e)>6))=0 for every positive S. (iii) urn Jwf(x : e) Q (dx) = 0. Proof Let D be the set of discontinuities off. As 40 the sets { x : wf(x : e) >6) decrease to a set Da . Now (i) means Q(D)=O and (ii) means Q(D8 )=0 for all 6>0. Since D= U D id , (i) and (ii) are equivalent. n>I Since, as 40, the functions wl( : e) are uniformly bounded and decrease to a function that is strictly positive on D and zero outside, (iii) is equivalent to Q(D)=0. Q.E.D. The next theorem provides two further characterizations of weak con- vergence of a sequence of probability measures. THEOREM 1.3. Let Q„ (n = 1, 2, ... ), Q be probability measures on a metric space S. The following statements are equivalent. (i) { Q„ } converges weakly to Q. (ii) lin m Q„(A)=Q(A) for every Borel set A that is a Q-continuity set. (iii) lim Jf dQ„ = Jf dQ for every complex-valued, bounded, Borel- measurable Q-continuous function f. Although it is not difficult to prove this theorem directly, we note that it follows as a very special case of Theorem 2.5. We conclude this section by recalling that if the metric space S is separable, then the weak topology on 9l is metrizable and separable, and that a metrization is provided by the Prokhorov distance dp between two probability measures Q, and Q2, which is defined by a,(Q1 ,Q2)=inf {E:c>O, Q1 (A)<Q2(A`)+e and Q2(A) < Q I (A`)+a for all Borel sets A ), (1.16) where A`={x:xES, p(x,A)<e}. (1.17)
- 33. 6 Weak Convergence and Uniformity Classes and p(x,A)=inf (p(x,y):yEA}. (1.18) For these and other details we refer the reader to Billingsley [1], (Appendix III, pp. 233-242) or Parthasarathy [1] (Chapter II, pp. 39-55). 2. UNIFORMITY CLASSES Unless otherwise specified, throughout this section S denotes a separable metric space. Let Q be a given probability measure on S. Our main task in this section is to characterize those classes f of complex-valued, bounded, Borel- measurable functions on S for which lim sup f fdQ„— f fdQI =0 (2.1) n fEc for every sequence of probability measures (Q„) converging weakly to Q. Such a class is called a Q-uniformity class of functions. A class 9 of Borel subsets of S is called a Q-uniformity class of sets if lim sup IQ(A)—Q(A)I=0(2.2) n AE(^ for every sequence of probability measures {Q„) converging weakly to Q. Thus (t' is a Q-uniformity class of sets if and only if = { IA : A E Q) is a Q-uniformity class of functions. We need some preparation before we can characterize uniformity classes. In the following lemma we deal with (signed) measures on a sigma-field of subsets of an abstract space I. For a finite signed measure on this space the variation norm II µII is again defined by (1.5) with 2 replacing S (and C replacing i3 ( ). LEMMA 2.1. (Scheffe's theorem) Let (St, LS ,A) be a measure space. Let Q„ (n = 1,2,...), Q be probability measures on (S2, in ) that are absolutely continuous with respect to A and have densities (i.e., Radon-Nikodym deriva- tives) q„(n = 1,2,...), q, respectively, with respect to A. If (q„) converges to q almost everywhere (A), then linmIIQ,,—Q11=O.
- 34. Uniformity Classes 7 Proof Let h„ = q — q. Clearly fh„dA=0 (n=1,2,...), so that IIQ, — QII= f h„da — f h„dX =2 f h„dx =2 f h„•!{ti,>o)dX. (2.3) (h„>0) (h„ G0) {h„>0) The last integrand in (2.3) is nonnegative and bounded above by q. Since it converges to zero almost everywhere, its integral converges to zero. Q.E.D. LEMMA 2.2. Let S be a separable metric space, and let Q be a probability measure on S. For every positive a there exists a countable family (Ak : k = 1,2,...) of pairwise disjoint Borel subsets of S such that (i) U (Ak : k = 1,2,...) = S, (ii) the diameter of Ak is less than a for every k, and (iii) every Ak is a Q-continuity set. Proof. For each x in S there are uncountably many balls (B (x : 8) : 0< 6 <e/2) (perhaps not all distinct). Since aB (x : 6) C { y : p(x,y) = S }, (2.4) the collection { aB (x : 3) : 0 < </2) is pairwise disjoint. But given any pairwise disjoint collection of Borel sets, those with positive Q-probability form a countable subcollection. Hence there exists a ball B (x) in { B (x : S) : 0 < S < e/2) whose boundary has zero Q-probability. The col- lection (B (x) : x E S) is an open cover of S and, by separability, admits a countable subcover { B (xk) : k = 1, 2, ... ), say. Now define A i =B(x1 ), A2=B(x2)B(x1),..., Ak=B(xk)(U_i'B(x1)),.... (2.5) Clearly each A k has diameter less than e. Since aA =a(SA), a(A n B)C(aA)U(aB), (2.6) for arbitrary subsets A, B of S. it follows that each Ak is a Q-continuity set. Q.E.D. To state the next lemma, we define, in addition to wf(• : e), the oscillation w^(A) of a complex-valued function f on a set A by wf (A) =sup {If(x)—f(y)I:x,yEA} (ACS). (2.7)
- 35. 8 Weak Convergence and Uniformity Classes Clearly, w1(x:E)= w1(B(x:E)) (xES, E>O). (2.8) LEMMA 2.3. Let f be a complex-valued, bounded, Bore/-measurable func- tion on a separable metric space S. For each pair of positive numbers E and 8 there exists a countable collection of pairwise disjoint Bore! sets (Nk : k = 1,2,...) such that (i) U (Nk : k = 1,2,...) D (x : w1(x : E) > S ), (ii) the diam- eter of Nk is less than 6E for every k, and (iii) w r(Nk)> S for every k. Proof. Let G= {x:wf(x:c)>S). Let (y:n=1,2,...) be dense in G, so that G C U (B (y E) : n =1,2,...). Let x, =y 1 ; let x2 be the first y„ whose n distance from x, is at least 2E; let x3 be the first y„ beyond x2 whose distance from (x1 ,x2 ) is at least 2E, and continue to get a countable set { x„ : n = 1,2, ...) such that (B (xn : E) : n = 1,2,...) is a pairwise disjoint collection. Also, since each y„ is within a distance of 2E from some xx, GCU {B(x,,:3E):n=1,2,...). Let B= U {B(x,,:c):n= 1,2,...) and define Mk =B(xk :3E)(B U U {B(xk,:3E):1 <k'<k)), Nk =B(xk:E)UMk. Note that the Mk 's are pairwise disjoint and U { Mk : k =1, 2, ...) J G B; also, Mk is disjoint from B. Hence (Nk : k = 1,2,...) is a pairwise disjoint collection of sets whose union contains G. Since Xk E G and B (xk : E) c Nk , it follows that wf(Nk) > wf(xk : E)> 8. Finally, Nk C B (xk : 3E), so that the diameter of Nk is not more than 6E. Q.E.D. We are now ready to prove the main theorem of this section. THEOREM 2.4. Let S be a separable metric space and Q a probability measure on it. A family F of complex-valued, bounded, Borel-measurable functions on S is a Q- uniformityclass if and only if (i) sup w1(S) < oo, In (ii) lim sup (Q ({ x : wf(x : E)>S))) = 0 for every positive 8. (2.9) fEJ Proof. The theorem is proved, without any essential loss of generality, for a class S of real-valued, bounded, Borel-measurable functions.
- 36. Uniformity Classes 9 SUFFICIENCY. Assume that (2.9) holds. Let c, a be two positive numbers. Let (Ak : k = 1,2,...) be a partition of S by Borel-measurable Q-continuity sets each of diameter less than e. Lemma 2.2 makes such a choice possible. Define the class of functions '+< , by f =1.Y.ckIA,:Ick lccforallk}. (2.10) k Then is a Q-uniformity class. To prove this, suppose that {Q„} is a sequence of probability measures converging weakly to Q. Define func- tions q,, (n= 1,2,...), q on the set of all positive integers by q.(k)=Q,,(Ak), q(k) = Q (Ak),(k=1,2,...). (2.11) The functions q, q are densities (i.e., Radon—Nikodym derivatives) of probability measures on the set of all positive integers (endowed with the sigma-field of all subsets) with respect to the counting measure that assigns mass one to each singleton. Since each A k is a Q-continuity set, {q(k)} converges to q(k) for every k. Hence, by Scheffe's theorem (Lemma 2.1), lim 2 Iq(k) — q(k)I= 0. (2.12) k Therefore sup (If fdQ^— f fdQl•f E `'` J =sup { t 12 ck q„(k)— 2 ck q(k) : Ick l < c for all k } k k JJJ ^ c2Iq„(k)—q(k)I-0 as n—oo. (2.13) k Thus we have shown that 9 ,, is a Q-uniformity class. We now assume, without loss of generality, that every function f in F is centered; that is, inf f (x) = — sup f (x), (2.14) xES xES by subtracting from each f the midpoint cf of its range. This is permissible because sup IJ fd(Q,, — Q)I = sup If (f—cf)d(Q„—Q)I (2.15) f ENf r=
- 37. 10 Weak Convergence and Uniformity Classes whatever the probability measure Q. For each f E define gJ(x) XEAkf(x) for xEAk(k=1,2,...), h1 (x)= sup f(x) for xEAk(k=1,2,...). (2.16) xEA k Note that g,h E 9 ^, gJ < f<hj, hf(x)—gf(x)=wf(Ak ) for xEAk(k=1,2,...). (2.17) It follows that f gjd(Q, — Q) — f (hJ — gJ) dQ = f gjdQ. — f hjdQ <f.fdQn — f fdQ<fhjdQ„ — f gjdQ = f h1d(Q, — Q)+ f (hJ — gj)dQ, (2.18) so that sup Iffd(Q„—Q)I< sup Iffd(Qn—Q)I+ sup f(hJ—gj)dQ. JE f J'E!^.. JE v (2.19) Since ^ is a Q-uniformity class, urn sup Iffd(Q.—C))I< sup f (hJ —gj)dQ n JEcJEeq = Sup I wf (Ak)Q (Ak) JE'F k 6 sup c Q(Ak)+ö JE J [ (wj(A.)>8 ) <csupQ(tx:(Jj(x:E)>S))+s, (2.20) JE J since the diameter of each Ak is less than e. First let 40 and then let 8j,0. This proves sufficiency.
- 38. Uniformity Classes 11 NECESSITY. Assume (2.9i) does not hold. Then there exists a sequence { f,,) c such that mJ (S)>n (n=1,2,...). (2.21) Thus if we write a„=inf(f„(x):xES}, Q=sup(f„(x):xES}, (2.22) then /3„ — a,, > n for all n. Divide the closed interval [a,,, /3J into n disjoint subintervals of equal length. There exists a subinterval I, such that Q(f^ I (I.,))>n (n=1,2,...). (2.23) Also, since /3,, — an > n, there exists a point x,, in S for which If„(x,,)—tl. n 2 for all tECI(I,,) (n=1,2,...). (2.24) Now define a probability measure Q„ by adding a point mass 1/n at x,, and subtracting this mass proportionately from subsets of f,^ '(I„); that is, Q.(A) = Q(Afn '(In))+ — (A) +f 1— nQ( f^ ^(I^)) ]Q(Anf„ '(I.)), (AE), (2.25) where Sx. is the probability measure degenerate at x„ (i.e., Sx. ({ x,, }) = 1). Note that IIQ.-Q11=n, (2.26) so that Q,, converges in variation norm and, therefore, weakly to Q. But f f dQ,, — Jf,,dQl = n II„(x,,)— Q f^ 1^ (I^) ✓J (i^)l,,dQ
- 39. 12 Weak Convergence and Uniformity Classes for some! in Cl(I„). Thus, by (2.26), f fn dQ. —f f" dQI >' (2.28) 2n for all n, implying that F is not a Q-uniformity class. Next assume (2.9ii) does not hold. This means that there exist positive numbers 8 and ri, a sequence {ç) of positive reals converging to zero, and a sequence (f) c 5 such that Q({x:w1 (x:f„) >6})>r1>0 (n=1,2,...). (2.29) Let (Nk. ,, : k = 1,2,...) be a countable collection of pairwise disjoint Borel sets satisfying (i) U( Nk,„: k=1,2,...)D(x:wJ (x:e„)>8), (ii) diameter of Nk.A < 6E„ for each k, (iii) wf (Nk. „)> 8 for each k (n=1,2,...). Such a collection exists (for each n) by Lemma 2.3. Given n, for each k choose two points xk, n, Yk, n in Nk „ such that f,, (Y) — f,, (x)> 8 (k= 1,2,...). (2.30) Thus T, Q( Nk,n)f (Yk,n) — 2 Q( Nk,,,)fn(xk,n) > s71, k k which implies that either I f f.dQ—YQ(Nk.,,)fl(xk.n)> (n=i,2,...), (2.31) k Nk.4 k or 7.Q(Nk.f)fn(Yk.f) -2 f fndQ> (n=1,2,...). (2.32) k k Nk., If (2.31) holds, then define Q„ by Q„(A)=Q(A U (Nk. .:k=1,2,... )) +2Q(Nk. .)S.,..(A) (A E ); (2.33) k
- 40. Uniformity Classes 13 if (2.32) holds, then define Q„ by Q(A)= Q(A U (Nk.f :k=1,2,... )) +2Q(Nk.,)8) (A) (A E (2.34) k Suppose (2.31) holds. Let f be a uniformly continuous complex-valued function on S. Then f fdQn— f fdQl =l Y. f Nk.. (f(xk.n)—f)dQl k wf(Nk.fl)Q(Nk.n) k <sup(wf (Nk,,,):k=1,2,... ). (2.35) The right side of (2.35) goes to zero as n goes to infinity, since the diameter of Nk. „ is, for all k, not more than 6e„ (which goes to zero). Hence {Q„) converges weakly to Q by Theorem 1.1. On the other hand f f.dQ — f fndQn=2(f f .dQ — Q(Nk.n)fn(xk.,,))> S 2 (n=1,2,...), k N,,, by (2.31), which shows that 5 is not a Q-uniformity class. A similar argument applies if (2.32) holds. Q.E.D. Remark. Let S be a separable metric space, with Q, and Q2 two probability measures on it such that Q2 is absolutely continuous with respect to Q 1 . It follows from Theorem A.3.1 (see Appendix), which characterizes absolute continuity, that every Q,-uniformity class of func- tions is also a Q2-uniformity class. The following variant of Theorem 2.4 is also useful. THEOREM 2.5. Let S be a separable metric space and Q a probability measure on it. A family 9 of complex-valued, bounded, Bore/-measurable functions on S is a Q- uniformity class if and only if (i) sup wf(S) < oo, In (ii) lim sup f w1(x : e)Q (dx) = 0. (2.36) CIO JE Proof. Suppose is a Q-uniformity class. By Theorem 2.4, (2.9) holds. Let c =sup {wj(S) : f E ). Given a positive 6 there exists a positive
- 41. 14 Weak Convergence and Uniformity Classes number Eo(S) such that sup Q I { x : wj(x : f) > ) / < 2c (2.37) J E g; `l for every E less than Eo(S). Hence for all f in + f wf(x:€)Q(dx)< f) {X:wf(x:t)< 21 4S whenever a is less than Eo(S). This proves necessity of (2.36). Conversely, suppose (2.36) holds. Choose and fix a positive number S. Given a positive, there exists a positive number E,(r^) such that sup f wj (x:c)Q(dx)<6r1 fE g i for all E less than E,(11). Hence for every f in f, Q({X:W1(X:E)>S})< -k f Wf(x:E)Q(dx)<77 for all a less than E,(ij). Thus (2.9) holds. Q.E.D. In order to specialize the above theorem to Q-uniformity classes of sets, we define A`= {x: p(x,A)<e} = U {B(x:e):xEA), A - `=(x:B(x:E)CA}=S(SA)`, (ACS, E>O). (2.38) The set A ` was also defined earlier in (1.16). Note that A ` is open and A -` is closed. COROLLARY 2.6. A class C?' of Bore! subsets of a separable metric space S is a Q-uniformity class if and only if lim sup Q(A`A - `)=0. (2.39) 110 A E a If every open ball of S is connected, then A`A - `=(aA)` (A CS, E>0), (2.40)
- 42. Uniformity Classes 15 so that in this case is a Q-uniformity class if and only if lim sup Q((aA)`)=0. (2.41) fio AE& Proof. For any arbitrary set A, wrA (x : e) is one if and only if B (x : e) intersects both A and SA; otherwise it has the value zero. Therefore W]A (X:E)=,A 'A -.(x) (XES, E>0). (2.42) Hence fwjA (x:e)Q(dx)=Q(A`A - `) (ACS, e>0). (2.43) Since w,A (S) < I for all sets A, it follows from Theorem 2.5 that (2.39) is a necessary and sufficient condition for & to be a Q-uniformity class. We now prove (2.40) under the hypothesis that every open ball of the metric space S [separability is not needed for the validity of (2.40)] is connected. First we show that the relation (aA)`CA`A - ` (ACS, e>0) (2.44) is valid in every metric space S. For suppose xE(aA)`. Then there exist a positive c' smaller than and a pointy in aA such that p(x,y) < e'. Since y is a boundary point of A, there exist two points z, and z2 , with z, in A and z2 in S A, such that p(y, z.) < E — e' for i = 1, 2. Thus p(x, z.) < p(x,y) + p(y, z; ) < e for i= 1, 2. This means that x E A ` and x A - `, which proves (2.44). Next we assume that every open ball of S is connected. If x E A ` A -`, then AnB(x:e)#4, (SA)nB(x:e)^. (2.45) We now suppose that xE(8A) (and derive a contradition). This means B(x':e)n 8A=0, (2.46) so that B(x:e)=((SCl(A))n B(x:E))u(Int(A)nB(x:e)), (2.47) since S = (S C 1(A )) U Int (A) U aA A. The right side of (2.47) is the disjoint union of two open sets. These two sets are nonempty because of (2.45), (2.46), and the relations (which hold in every topological space) (SC1(A))U aA D SA, Int(A)U aA DA.
- 43. 16 Weak Convergence and Uniformity Classes However this would imply that B (x : e) is not connected. We have reached a contradiction. Hence xE(8A)`, and (2.40) is proved. The relation (2.41) is therefore equivalent to (2.39). Q.E.D. COROLLARY 2.7. Let S be a separable metric space. A class of bounded functions is a Q-uniformity class for every probability measure Q on S if and only if (i) sup {wt(S) : f E } < co, and (ii) is equicontinuous at every point of S; that is, lim sup w1(x : () = 0 for all x E S. (2.48) JE J Proof We assume, without loss of generality, that the functions in 15 are real-valued. Suppose that (i) and (ii) hold. Let Q be a probability measure on S. Whatever the positive numbers S and a are, sup Q{{x:,1(x:e)>6))<Q({x: sup wf(x:e)>S}).t (2.49) JE 9 J fE^ For every positive 8 the right side goes to zero as €,0. Therefore, by Theorem 2.4, is a Q-uniformity class. Necessity of (i) also follows immediately from Theorem 2.4. To prove the necessity of (ii), assume that there exist a positive number S and a point xo in S such that sup c j (xo :e)>S foralle>0. fE9 This implies the existence of a sequence {x„) of points in S converging to xo and a sequence of functions (f„) c F such that IL(xn)—f.('XO)I>2 (n=1,2,...). Let Q = Sxo and Q„ = Sx. (n =1, 2, ... ). Clearly, (Q} converges weakly to Q, but ff.dQ,^ — f.dQl =If^(x,^)—f,,(xo)l>2 (n=1,2,...). Hence is not a Q-uniformity class. Q.E.D. We have previously remarked that the weak topology on the set Vii' of all probability measures on a separable metric space is metrizable, and the tThe set (x: sup(wj(x : c) : f E f) >6) = u ((x : wj(x : c) > 8):f E i} is open (see Section 11) and, therefore, measurable.
- 44. Uniformity Classes 17 Prokhorov distance dp metrizes it. Another interesting metrization is pro- vided by the next corollary. For every pair of positive numbers c. d, define a class L(c, d) of Lipschitzian functions on S by L(c,d) = { f : wf (S) < c, I f (x) — f (y)l < dp(x,y) for all x,y ES). (2.50) Now define the bounded Lipschitzian distance dBL by dBL (Q1 , Q2) = sup I fdQ, — f fdQ2I (Q1.Q2E''P). (2.51) fEL(1,1) COROLLARY 2.8. If S is a separable metric space, then dBL metrizes the weak topology on Proof. By Corollary 2.7, L(l, I) is a Q-uniformity class for every probabil- ity measure Q on S. Hence if (Qn ) is a sequence *of probability measures converging weakly to Q, then lira dBL (Qn,Q)=0. (2.52) n Conversely, suppose (2.52) holds. We shall show that (Q n ) converges weakly to Q. It follows from (2.52) that lim l f fdQn — f fdQl =0 for every bounded Lipschitzian function f. (2.53) For, if f (x) — f (y)^ < dp(x,y) for all x,y E S, f fdQ,,— f fdQ=c( f f'dQn — f f'dQ), where c = max {wf(S),d) and f' = f/c E L(1, 1). Let F be any nonempty closed subset of S. We now prove lim Qn (F)<Q(F). (2.54) n For E >0 define the real-valued function fE on S by ff (x)=^(e - 'p(x,F)) (x ES), (2.55)
- 45. 18 Weak Convergence and Uniformity Classes where 4, is defined on [0, oo) by (t) __1. 1—t if 0 < t < 1, (2.56) 0 if 1>1. Note that f is, for every positive e, a bounded Lipschitzian function satisfying wf(S)< 1,If,(X) — f^(Y)I<) EP(x+F) — - p(Y,F)I EP(x,Y) (x, Y E S ), so that, by (2.53), linm f f^ dQ„ = f f, dQ (e>0). (2.57) Since IF f, for every positive e, lin m Q,, (F) < lim f f dQ„ = f f dQ (e>0).(2.58) Also, lim f^(x)= IF(x) for all x in S. Hence lim JJdQ=Q(F). (2.59) cj0 By Theorem 1.1 { Q) converges weakly to Q. Finally, it is easy to check that dBL is a distance function on 'P. Q.E.D. Remark. The distance daL may be defined on the class GR, of all finite signed measures on S by letting Qt . Q2 be finite signed measures in (2.51). The function µ—.d,L (s,0) is a norm on the vector space 9R.. The topology induced by this norm is, in general, weaker than the one induced by the variation norm (1.5). It should also be pointed out that the proof of Corollary 2.8 presupposes metrizability of the weak topology on 9 and merely provides a suitable metric as an alternative to the Prokhorov distance do defined by (1.16)]. This justifies the use of sequences (rather than nets) in the proof. One can construct many interesting examples of uniformity classes beyond those provided by Corollaries 2.7 and 2.8. We give one example now. The rest of this section will be devoted to another example (Theorem 2.11) of considerable interest from our point of view.
- 46. Uniformity Classes 19 Example Let S=R 2. Let 6! (1) be the class of all Borel-measurable subsets of R 2 each having a boundary contained in some rectifiable curvet of length not exceeding a given positive number 1. We now show that 6D (I) is a Q-uniformity class for every probability measure Q that is absolutely continuous with respect to the Lebesgue measure A 2 on R 2. Let A E 6 (l) and let 3A C J, where J is a rectifiable curve of length 1. There exist k points zo,z t ,.••,zk -, on J such that (i) zo and zk _ t are the end points of J (may coincide), (ii) k<I/c+2, and (iii) Iiz1 —z1 _,II<E for i=1,2,...,k-1 (11.11 denotes euclidean norm). The k open balls having z;'s as centers and radii 2E cover J`. Hence A2((aA)`) <1A2(J (L +2)1T(2E) 2 =41rk+87rE 2[A E (l )]. (2.60) Let Q be a probability measure that is absolutely continuous with respect to A2. Then (2.60) implies (in view of Theorem A.3.1) lim (sup{Q((aA):AE^'i)(1)})=0. (2.61) By Corollary 2.6, -D (1) is a Q-uniformity class. We need some preparation before proving the next theorem. Let S be a metric space. Define the Hausdorff distance A between two closed bounded subsets A, B of S by A(A,B)=inf{ E:c>0,AcB`, BCA`}. (2.62) The class Q of all closed bounded subsets of S is a metric space with metric A. LEMMA 2.9. Let ')1t be a compact subset of (? . Then for every probabil- ity measure Q on S one has lim sup{Q(M`):ME )=sup{Q(M):MElt). (2.63) cjo Proof Let rq be a given positive number. For every M E )lt. there exists a positive number E,t, such that Q(M`")<Q(M)+"1 , to rectifiable curve in R 2 is a subset of R 2 of the form {z(t):0G t < 1), where t- z(t) =(x(t),y(t)) is a continuous function of bounded variation on [0,11 into R 2 ; z(0), z(1) are called the endpoints of the curve.
- 47. 20 Weak Convergence and Uniformity Classes since M`J.M as 40. By compactness of Olt, there exists a finite collection of sets {M1 , ..., M,) in OL such that every set in OJlt, is within a A-distance from M for some i, 1 <i < r. Let EM; Eo =-min { :1<i<r } . Then sup(Q(M`"):MEOL) <sup{Q(M,`°Mi2):1<i<r} <sup(Q(MiAli ): 1 <i <r) <sup {Q(Mi )+rl: 1 < i < r} <sup (Q(M): M E Olt,}+rl. Q.E.D. A subset C of Rk is convex if a x, + (1 — a) x2 E C for all x,, x2 E C and all a E [0,1]. A hyperplane H in R' is a set of the form H={x:<u,x>=c}, (2.64) where c is a real number and u is a unit vector of R k ; that is, Ilul) = 1, and <,> denotes euclidean inner product k <u,X>= u,X,, (2.65) i=! k 1/2 (lull u? [u=(u1,...,uk), x=(x 1 ,...,xk )ER k ]. i^l A closed half space E is a set of the form E=(x:xER k ,<u,x)<c) {cER', uER k , IIull=l]. (2.66) A hyperplane H= (y : <u,y> = c) is said to be a supporting hyperplane for a set A at x E A if <u,x>=c, AC{z:<u,z><c}, (2.67) that is, if the hyperplane H passes through the point x of A and has A on one side. Note that the sign < in (2.66) and (2.67) may be replaced by >
- 48. Uniformity Classes 21 (by merely changing u to — u). It is a well-known fact that if A is a compact convex set, then there exists a supporting hyperplane for A at each x E aA .t LEMMA 2.10. Let A, B be two closed, bounded, convex subsets of R k . Then A(A,B)=A(aA,aB). (2.68) Proof. Suppose A(8A,aB) =E. Let E' be any number larger than E. Let x E A. There exist xr, x2 E aA and a E [0,11 such that x = ax I +(1 — a)x2, since the intersection of A with any line through x is a closed line segment whose end points (x,,x2, for example) lie on 8A. Let y,, y2 E aB be such that xi —y; < E' for i= 1, 2. Then letting y = ay, +(l — a)y2 yields y E B and llx — yl! <«Ilx1 — yell+(I — a)IIx2 — y2fl <E'. Thus A c B` . Similarly B CA '. Since this is true for every E > E, 0(A, B) < A(aA, aB ). (2.69) To prove the opposite inequality, suppose that E > t1(A, B) and let be any positive number. Let x E aA A. Let (z : (1, z> = c) be a supporting hyperplane for A at x. Then the half space H = { z : (l, z> c c + E) contains A. If z E A and 11z' — z l l< E, then (l,z'>=(l,z>+(l,z'—z> <l,z>+IIz'—z!I c+ e. Hence H J A D B. The ball B (x : E + il) intersects R k H. This is because the point x +(E +71/2)1 of this ball satisfies (l,X+(E+ 2)l>=(l,x>+( E+ 2) =C +E+ 2 , and therefore lies in the complement of H. It follows that B (x : E + rl) intersects R k B. But, since A C B ` and x E A, B (x : E + rl) certainly inter- sects B. It follows that B (x : E + rl) intersects aB, so that (aB)"'' D 3A. Similarly (8A)` + ' j B. Therefore A(aA,aB) <E +rl tEggleston II), p. 20.
- 49. 22 Weak Convergence and Uniformity Classes for every c > 0(A, B) and every positive rt. Hence 0(aA,aB) <0(A,B). (2.70) The inequalities (2.69) and (2.70) together yield (2.68). Q.E.D. THEOREM 1.11. Let C denote the class of all Bore/-measurable convex subsets of Rk. Let Q be a probability measure on R k. The class e is a Q-uniformity class if and only if it is a Q-continuity class, that is, if and only if Q(ac)=o forallCEe. (2.71) Proof If e is a Q-uniformity class, then, by relation (2.41) in Corollary 2.6, e is a Q-continuity class. Suppose, conversely, that (2.71) holds. Since the characterization (2.41) of Q-uniformity is in terms of boundaries, and since aC= a (Cl(C)) for every convex set [note that C1(C)= Int(C)u aC and that aC has empty interior for convex C], it follows that C is a Q-uniformity class if and only if e = (C 1(C) : CE C) is. Given ri >0, let r be so chosen that Q({x:IIxII>r))<2. (2.72) Write C', = { C: C E e, CCC1(B (0: r))). By a well-known theorem of Blaschket 3, is compact in the Hausdorff metric A defined by (2.62). Lemma 2.10 shows that the compactness of , is equivalent to the compactness of { aC : C E 13, }. Lemma 2.9 and the hypothesis (2.71) now yield lim sup{ Q((8C)`):CEC',}=0. (2.73) By Corollary 2.6, C, is a Q-uniformity class. Let (Qn ) be a sequence of probability measures weakly converging to Q. Then lim (sup{IQn(C) — Q(C)I:CEC^)) n <11im (sup{IQ.(C)—Q(C)^:CE,}) + lim Qn({x:llxll>r))+Q({x:llxll>r)) n =2Q({x:fIxll >r})<ii. (2.74) tSee Eggleston [1), pp. 34-67.
- 50. Integrals Over Convex Shells 23 Note that the last equality follows from the fact that (x : Ilxil > r) is a Q-continuity set, since its complement is [although, given any probability measure Q and a positive rl, one can always find r such that (x: jlxjl > r) is a Q-continuity set and (2.72) holds]. Since rl is an arbitrary positive number, it follows from (2.74) that a is a Q-uniformity class. Con- sequently, (2 is a Q-uniformity class. Q.E.D. Remark. It follows from the above theorem that if Q is absolutely continuous with respect to Lebesgue measure on R", then e is a Q-uniformity class. In particular, a is a 4)- uniformity class, 4) being the standard normal distribution in R k . We shall obtain a refinement of this last statement in the next section. It is easy to see from the proof of Theorem 2.11 that the class C in its statement may be replaced by any class Ef of Borel-measurable convex sets with the property that R,- (CI(A)nCl(B(0:r)):AE($} is compact in the Hausdorff metric for all positive r. In particular, by letting d — {(—oo,x1]X(—oo,x2]X ... X( — oo,xk]:x—(xi,...,Xk)ER k }, we get Polya's result: Let P be a probability measure on Rk whose distribution function F, defined by F(x)-P((-oo,x,]x... x(-oo,xk ]) [x-(x......xk)ERk ], (2.75) is continuous on Rk. If a sequence of probability measures {P„} with distribution functions {FA ) converges weakly to P, then sup(IF„(x)-F(x)I:xER k }-+0 (2.76) as n-boo. The left side of (2.76) is sometimes called the Kolmogorov distance between Po and P. The converse of this result is also true: if (2.76) holds, then (P„) converges weakly to P. In fact, if (F„} converges to Fat all points of continuity of F, then (P„) converges weakly to P.t 3. INEQUALITIES FOR INTEGRALS OVER CONVEX SHELLS It is not difficult to check that if C is convex then so are Int(C),C1(C). The convex hull c(B) of a subset B of R k is the intersection of all convex sets containing B. Clearly c(B) is convex. If C is a closed and bounded convex subset of Rk, then it is the convex hull of its boundary; that is, c(8C)=C. Clearly c(8C) C C. On the other hand, if x E C, x li= 8C, then every line through x intersects aC at two points and x is a convex combination of these two points. Thus c(aC) = C. If C is convex and e >0, then C` is tSee Billingsley [ 1 1, pp. 17-18.
- 51. 24 Weak Convergence and Uniformity Classes convex and open, and C -' is convex and closed. The main theorem of this section is the following: THEOREM 3.1. Let g be a nonnegative differentiable function on [0, oo) such that (i) b= f Ig'(t)It k- 'dt<cc, 0 (ii) lim g(t)=0. r--► oo Then for every convex subset C of Rk and every pair of positive numbers e, p, g(JI xI1)dx < bak(E+p) (3.1) C'C - v where k7l k/2 2^7k/2 ak _ f((k+2)/2) I'(k/2) ' (3.2) is the surface area of the unit sphere in Rk. COROLLARY 3.2. Let s > 0, k > 1, and f(x)=(21T)-k/2IIxIIsexp j — 11x112 1 (xERk ) Then for all convex subsets C of Rk and every pair of positive numbers e, p, f c: '/2(2s r((k+s-1 )/2) f(x)dx<2 -+k-1) r(k/2) ((+p). (3.3) C.C -, Proof. Here one takes (in Theorem 3.1) g(t)=(217) -k/2 t'exp(-12/2) 1E[0,00). Then g,(t)=(2r)-k/2(st$-I— is+ I )exp (— 2 1, ( )—k/2 f o 00 ( k+s-2 k+s) p r — 2 l di b < 2^r st + t ex j 2 } l J
- 52. Integrals Over Convex Shells 25 =(2.r)-k/2(s,2(k+s-3)/zr( k+s— 1 )+21c+3_/2r( k+s+ 1 2 2 k/2 (k+s-3)/2 =(2^r) - 2 (2s+k-1) r( k+s - 1 2 =2(s -3)/2(2s+k— I) -k/2 r( k+s-1 ) (3.4) which gives (3.3) on substitution in (3.1). Q.E.D. The rest of this section is devoted to the development of the material needed for the proof of Theorem 3.1. For k=1, C` C -° is contained in the union of two disjoint intervals each of length e+ p. Hence f g(IxI)dx 2(e +)( sup g(x)) <2(e+p) f 'Q I g'(t)I dt=2(e+p)b C`C - ^ x>0 0 =ba 1 (e+p). (3.5) For k >. I a more intricate argument is needed. For the rest of the section we assume k> 1. A polyhedron is a closed, bounded, convex set with nonempty interior that is the intersection of a finite number of closed half spaces. If P is a polyhedron, a face of P is a set of the form H n aP, where H is a hyperplane such that H n aP has nonempty interior in H. LEMMA 3.3. Let a polyhedron P be given by P=(x:<u^,x)<d^, 1 < j<m}, (3.6) where ui 's are distinct unit vectors. Let Li= (x:xEP, <u^,x>=d) (I < j<m). (3.7) Then a P = U L^. If F is a face of P, then F = L^ for some j. Moreover, P= (x : (u^,x> < d for all j for which LL is a face of P }. (3.8) Proof. The first assertion is obvious. If F= H n 8P= u(H n L^) is a face of P, then H = (x : <u^, x) = d) = H^, say, for exactly one j, since the intersection of H with any hyperplane distinct from H has empty interior in H. This proves the second assertion. For the third, note that
- 53. 26 Weak Convergence and Uniformity Classes Lj = Hj n aP. It is clear that the interior of L1 in H^ is (x:<u,,x><d, for r#j, <u,,x>=d ), so that if Lj is not a face of P, then Lj C U L,. This implies 8P C 8Q, r#j where Q is defined by Q= { x : <uj,x> <d for allj for which L1 is a face of P). Clearly P c Q. It is sufficient to show that aP c aQ implies P= Q. Let xt E Int (P). Assume there exists x 2 E aQ aP. Consider the line segment [x1 ,x21 joining x, and x2. This line segment meets aP at x 3, say. Clearly, [x1 ,x2)CInt(Q) and x3 x2, so that x3 EInt(Q)naP, which contradicts the fact aP c Q. Q.E.D. LEMMA 3.4. A polyhedron P has a finite number of faces. if F1 ,.. .,Fm are the faces of P, then a P = U F. Moreover, there exist unique unit vectors u^ and constants d3 such that F, = (x : x E P, <u^, x> = dd ), and P C (x : <u^, x> < di ). Also, P then has the representation P=(x:<uj,x><dj, l <j<m}. (3.9) Proof. This lemma follows easily from Lemma 3.3. Note that the repre- sentation (3.9) is unique up to a permutation of unit vectors uj's and the corresponding constants d's. Q.E.D. Remark. A polyhedron is the convex hull of a finite number of points. In fact, since each face is a polyhedron in a lower dimensional affine space, this follows by induction on k. Conversely, it is known that the convex hull of a finite set of points can be expressed as the intersection of a finite number of closed half spaces.t There are two main steps in the proof of Theorem 3.1. The first one (Lemma 3.9) is to express the surface integral as a derivative of volume in volume integrals. The second step (Lemma 3.10) is to get a uniform bound for the surface integral of a fixed function over the boundary of a polyhedron. The ideas involved here belong naturally to the domain of surface area and surface integrals. In the following paragraphs we develop the material needed for the proofs. The development here is self-contained except for the use of Cauchy's formula, which is stated but not proved. We begin with some notation. Let Ak denote the Lebesgue measure on R k normalized by the euclidean distance on R k ; that is, if y 1 , ... ,yk is an tSee Eggleston 111, pp. 29-30.
- 54. Integrals Over Convex Shells 27 orthonormal basis and A is a cube with respect to them or A = { I ty; : a; < t; < b; for all i}, then X.(A)=(bt — at )• • • (bk — ak). We also refer to Ak as the k-dimensional Lebesgue measure on R'. On any hyperplane of R' there is a (k — 1)-dimensional Lebesgue measure normalized the same way. We denote this measure as Ak _ 1 and call it the (k— 1)-dimensional Lebesgue measure. For example, if H is a hyperplane, it can be written in the form H=xo +Ry 1 +---+Ryk _ 1 , where x0,y1,...,yk-1EH and y,,•..,yk _ I are (k—I) orthonormal vectors in R'. If f is a function with compact support in R k, then fHfdXk-1= f f(xo+t1Y1+ ... +tk-1Yk-1)dt,...dtk_1. (3.10) Next we denote ak _ l as the surface area measure on the unit sphere Sk _ . One can write an explicit formula for 0k- 1 by using Eulerian angles to parametrize points of Sk _ 1 . It then follows that ok _ I (Sk _ 1 )=ak , where a, is given by (3.2). Let P be a polyhedron with faces F, (1 < i < m). Then the surface area of P is defined as Ak_1(8P)= Ak-1(F;). (3.11) ;=t For a bounded Borel-measurable function f on R k we define the surface integral off on a P by faPfdXk-1= ^, f fd^k-1 (3.12) and if A is a Borel subset of aP, then the surface area of A is defined by Ak-1(A)= Ak-1(AnF;). (3.13) Remark. Let P be given as P={x:<u^,x><4., I <j< m}. Then f fdak _ 1 = f fdAk_1. (3.14) aP I<j<m L, where L L -(x:xEP,<u1,x>=d 1 ). This follows from the fact that A1(L)0 if L^ is not a face of P.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. Betaught, [bi-tahte] == taught. Rel. S. v. 124 Bete, v. a. lit. == ‘make better;’ hence, ‘heal,’ ‘save.’ Marg. 68 —— == ‘recompense,’ ‘make amends for.’ RG. 369. AS. bétan Bete, part. == beaten. Vid. Beat Beten, [y-beten] == overlaid, covered, as with silk, gold, &c. Alys. 1034, 1518 Beth, Beoth, &c. See Be Bethink, v. a. == ‘to bethink oneself’ of a thing. RG. 368, 458 Betide, v. n. == happen. RG. 418, 14 Betime, adv. K. Horn, 995 Betoken, v. a. RG. 152 Betokening, sb. RG. 560 Betray, v. a. RG. 135 Better, adj. RG. 367, 422 —— v. n. == get the better. Ps xii. 5 Betterness, sb. Ps. li. 5 Between, prep. RG. 371, 513 Betwixt, prep. [bi-tuxen]. O. and N. 1745 Beverage, sb. == drink. RG. 26 —— == reward, consequence. RG. 299 Bewail, v. a. Alys. 4395 Beware, v. n. RG. 547
- 57. Beweep, v. a. O. and N. 972 Bewind, v. a. == entwine. part. ‘bewound.’ Christ on the Cross, 3 Bewray, v. a. == betray [by-wrye]. Alys. 4377. pret. ‘bi-wro.’ O. and N. 673. AS. wrégan. Beyen, == are? Wright’s L. P. p. 32 Beyond, prep. RG. 368, 420 Beyre, == of both, gen. pl. RG. 388, 398 Bezant, sb. == a piece of money. RG. 409. From Byzantium, or Constantinople, where they were originally used Bible, sb. Rel. Ant. ii. p. 174 Bicast, v. a. == cast over, cover. 92 β Bicatch, v. a. == deceive, ensnare. Alys. 258. K. Horn, 318 Bicharred, part. == deceived. Rel. Ant. ii. p. 211; M. Ode, 160. AS. becýrran Bicherme, v. a. == chirp about or around. O. and N. 279. AS. cyrm Bick, v. n. == fight. Alys. 2337 Bicker, v. n. == quarrel. RG. 540. Fr. becquer. W. bicra == to fight Bicker, sb. == a quarrel, contention, battle. RG. 538, 543 Biclipe, Biclupe, v. a. == accuse. 365 B. —— == appeal. RG. 473 Biclose, v. a. == enclose. RG. 558, 218 Bid, v. a. == ask. RG. 77. 3 pl. pret. ‘badden.’ Alys. 5823. See ‘bede’ —— == command. RG. 29. pret. ‘bad.’ 683 B. part. ‘y-bede.’ RG. 383. AS. biddan
- 58. Bid, v. a. == offer, pret. ‘bode.’ RG. 379. ‘beod’? O. and N. 1435. AS. beódan Bid, sb. == asking, demand. Pol. S. 149 Bidding, sb. == demand, request. Pol. S. 150 Bide, v. n. == remain. Pol. S. 204 Bidene, adv. == presently. Ps. l. 4; ciii. 30 Bidelve, v. a. == bury. Rel. Ant. i. 116 Bidone, part. == ‘bidun in grave.’ Body and Soul, 97 Bier, sb. 128 B. Bieren, sb. == a man. Ps. cxxvi. 5; cxxxix. 2. AS. beorn Biflette, v. n. == flow past. K. Horn, 1457 Bifluen, v. a. == flee from. M. Ode, 77 Big, v. a. == build. Ps. xxvii. 5. AS. byggen. ON. byggja Bigabbed, part. == deceived. Lit. ‘talked over.’ RG. 458. AS. gabban Bigate, sb. == booty. Alys. 2138 Biggand, sb. == a builder. Ps. cxvii. 22 Biglide, v. n. Wright’s L. P. p. 87 Bigrede, v. a. == lament. Alys. 5175. AS. grædan. —— == call to. O. and N. 279 Bihaite, v. a. == behold? O. and N. 1320. AS. behawian. Or, possibly, == observe, regard. AS. hedan. Germ, behüten. See Gloss. Rem. on Laȝ. iii. 457 Bihalves, adv. == aside. St Kath. 13
- 59. Bihede, v. a. == regard. O. and N. 635 Bihemmen, v. a. == cover, cloak. O. and N. 672 Bihepe, part. == heaped up. O. and N. 360 Bihete, v. a. == promise, pret. ‘bihet.’ RG. 381. ‘byheyghte.’ Alys. 3926 Bihoting, sb. == promise. Alys. 4000 Bike, sb. == cassia. Ps. xliv. 9. Literally ‘pitch.’ ON. bik Bilace, part. == beset. Alys. 3357 Bilaue. See Bileve Bilaucte. See Bilou Bilede, v. a. == lead about. Pol. S. 155. O. and N. 68 Bilegge, v. a. == assert, allege. O. and N. 672 Bileve, v. a. == leave. RG. 421 —— v. n. == remain. RG. 372, 374. [bilaue]. Alys. 3541 Biliked, part. == rendered likely or probable. O. and N. 840 Bilime, v. a. == to mutilate. RG. 471, 560 Bilimp, v. n. == happen. M. Ode, st. 59 AS. belimpan Bill, sb. (of a bird). O. and N. 79 —— == hatchet. Pol. S. 151 Bilou, pret. == laughed at. RG. 328. [bylaucte]. K. Horn, 681. [by lowe], RG. 299. [by lowȝ]. RG. 64 Bimong, prep. == among. Wright’s L. P. p. 35 Bind, v. a. Wright’s L. P. p. 45. part. ‘ibounde.’ RG. 487
- 60. Binder, sb. HD. 2050 Binding, sb. == chain. Ps. cxxiv. 5 Bink, sb. See Bench Bipahte, pret. == deceived. Rel. S. v. 128. AS. be-pæcan Birade, v. a. == counsel. Alys. 3732 Birch, sb. == the tree. Alys. 5242 Bird, sb. RG. 177 Birde, sb. == lady. HD. 2760. A metathesis of ‘bride’ Birst, == bruised. Body and Soul, 86. AS. berstan. Birth, sb. == nation. Ps. lxxviii. 10 Birthman, sb. == man of good birth. HD. 2101 Birthtime, sb. [burtyme]. RG. 9, 443 Birue, v. a. == rue, repent. Fragm. Sci. 325 Bis, sb. == purple. Wright’s L. P. p. 26. Fr. bis. Lat. byssus Bisay, v. a. == recommend, say. RG. 422 Bisayen, == treated. See Besee. Bischriche, v. a. == shriek at. O. and N. 67 Biscunien, v. a. == shun. M. Ode, 77 Bise, sb. == north wind. HD. 724. OHG. bísa Bisend, v. a. == send after. RG. 491 Bishop, sb. RG. 376 Bishopric. RG. 414, 417
- 61. Bismere, sb. == blasphemy. Body and Soul, 110. [busemere]. RG. 12, 379. AS. bismér Bisne, adj. == blind. O. and N. 78. AS. bisen Bisoht, == sought out, got ready for. Pol. S. 220 Bisokne, sb. == beseeching. RG. 495 Bispel, sb. == proverb. O. and N. 127. AS. bispel Bistad, sb. == a dwelling. Wright’s L. P. p. 38 Bistand, v. a. == stand by a person; hence, to press or urge them. O. and N. 1436 Bistolen, part. == stolen, crept onwards. M. Ode, 9 Bisyhed, == the state of being busy. Alys. 3 Bit, sb. == a morsel. RG. 207 Bit, sb. == bottle. Ps. lxxvii. 13. [bite]. Body and Soul, 34. AS. bitte Bitch, sb. Alys. 5394 Bite, v. a. Alys. 5435 Bite, sb. Alys. 5436 Bite, v. a. == drink. HD. 1731. Cf. bohem. ‘piti,’ potus; ‘pitka,’ potatio, &c. Gr. πίνω Bitell, v. a. == excuse. O. and N. 263 Bitiȝt, == arrayed. O. and N. 1011. AS. biþæht. See Gloss. to Laȝ. s. v. Bitoȝe, == employed. O. and N. 702. AS. biteon. See Gloss. to Laȝ. s. v. Bitter, adj. Wright’s L. P. p. 87
- 62. Biturn, v. a. == turn. RG. 210 Bituxen. See Betwixt. Biwene, v. a. == discover, recognize. O. and N. 1507 Biwente, vb.—‘hire bi-wente.’ == turned her about. K. Horn, 329. In pass. ‘þai bewent’ == let them be turned back. Ps. vi. 11. AS. wendan Biwere, v. a. == protect. O. and N. 1124. AS. bewerian Biweved, == covered. RG. 338. —— == woven? Alys. 1085 Biwin, v. a. == win. RG. 75, 420 Biwit, adv. == out of one’s wits. RG. 528 Biwite, v. a. == defend. Rel. S. v. 252. AS. bewitan —— == know? Alys. 5203 Biwrye, v. a. == cover. Alys. 6453. AS. wreon. Black, adj. RG. 433, 522 Blacken, v. n. == become angry. HD. 2165 Blame, v. a. RG. 163 —— sb. RG 272, 432 Blandishing, sb. == blandishments. St Kath. 164 Blanis ? Alys. 6292 Blanket, sb. 1167 B. Fr. blanchet Blast, v. n. == blow, puff. Alys. 5349 Blast, sb. Fragm. Sci. 190. Ps. cxlviii. 8
- 63. Blaze, sb. 1254 HD. AS. blǽse, blýsan Blear, v. n. == become bleareyed. Rel. Ant. ii. p. 211 Bleat, v. n. Ps. lxiv. 14 Bled, blete, sb. == foliage. O. and N. 1040, 57. AS. blæd Bleed, v. n. RG. 560 Bleike, adj. == pale. HD. 470. AS. blác. ON. bleikr Blench, sb. == a trick? O. and N. 378. ON. blekkja Blench, v. n. == avoid (a thing). O. and N. 170 —— == flinch from [blinche]. 2184 B. —— == deceive. Ritson’s AS. viii. 23 —— == give way? (of a ship) K. Horn, 1461. Another form of ‘flinch.’ AS. blinnan Bleo, sb. == hue, complexion. O. and N. 152. Wright’s L. P. p. 35. AS. bleo Bless, v. a. RG. 406 Blessing, sb. RG. 421 Blete, adj. == bleak? O. and N. 616 Blete, sb. See Bled Blike, v. n. == shine. Wright’s L. P. p. 52 AS. blícan Blinch. See Blench Blind, adj. RG. 376, 407 —— v. n. == become blind. Rel. Ant, ii. p. 211
- 64. Blink, sb. ‘to make blinks,’ == deride a person. HD. 307. See Blench, sb. Blin, v. n. == cease. RG. 566. pret. ‘blenyte.’ RG. 338. AS. blinnan Bliss, sb. RG. 469 Blissful, adj. Wright’s L. P. p. 52 Blissfully, adv. Ps. xcvi. 1 Blithe, adj. RG. 15 Blithely, adv. 89 β Blitheful. Ps. cxi. 5 Blive, adv. == quickly. RG. 544. See Belive Blode, adj. == pale, dried up. Rel. Ant ii. p. 210. Germ. blöde Blood, sb. RG. 388, 416 Bloody, adj. RG. 304, 311 Bloom, sb. HD 63 Bloom, v. n. Ps. xxvii. 7 Blote, adj. == dried. Rel. Ant. ii. p 176 Bloute, v. n. == swell out? HD. 1910. ON. blautr. Eng. bloat Blow, v. a. == as ‘blow the fire.’ HD. 385. Alys. 5030 —— v. n. pret. ‘blew.’ 524 β Blow, vb. n. part. ‘blowe,’ == blown, in blossom. O. and N. 1634 Blowing, sb. 467 β Blue, adj. [blo]. Wright’s L. P. p. 86
- 65. Bo, == be. O. and N. 166, et passim. See Be Bo, == both. q. v. Boar, sb. RG. 133 Board, sb. == table. 122 β.; plank. Alys. 6415 Boast, sb. RG. 258. pomp. St Swithin, 43 Boast, v. n. Alys. 2597 Boasty, adj. == boastful. Fragm. Sci. 283 Bobance, sb. == boasting. Pol. S. 189 Fr. bobance Bochevampe, (sic in MS.). == botched vamps or fronts of shoes. Rel. Ant. ii. p. 176 Bode, sb. == commandment. Ps. cxviii. 134, 128, et passim Bode, v. a. == foretell. O. and N. 530 Boded ? Pol. S. 152 Boding, sb. RG. 416, 428 Bodeword, sb. == message. Ps. ii. 6 Body, sb. RG. 395, 547 Boffing, == swelling or puffing. RG. 414. Fr. buffer, to puff the cheeks Boistous, adj. == coarse, rude. Alys. 5660. [boustes] Fragm. Sci. 273 Bold, sb. == a building. RG. 44. AS. bold Bold, v. a. == embolden. Alys. 2468. [bald]. Ritson’s AS. viii. 128 —— adj. RG. 383. ‘bolder.’ RG. 465
- 66. Boldhede, == boldness. O. and N. 514 Boldly, adv. RG. 500, 19 Boleax, sb. == large axe. Rel. Ant. ii. p. 176. ON. bolöxi Bolken, v. n. == belch. Ps. cxliii. 13 Bollen, == swollen. Body and Soul, 31. ‘ibolȝe.’ O. and N. 145 Bolster, sb. Rel. S. v. 90 Bolt, sb. ‘ȝoure bolt is sone ischote.’ St Kath. 54 Bonde, sb. == bondman. Pol. S. 150 Bondman. RG. 370. HD. 32 Bone, sb. == os. RG. 446 Bone, sb. == prayer. RG. 14. AS. bén. SS. bone Boned, [y-boned] == having bones. RG. 414 Bonére, adj. == debonair, graceful. Alys. 6732 Bonny, adj. Alys. 3903 Book, sb. RG. 374, 420 Boot, sb. == use, avail. Body and Soul, 92 —— == remedy, means (bote). RG. 277, 408. Pilate, 139 Booth, sb. Alys. 3457 Booze, sb. [bous] == drink. Wright’s L. P. p. 111. Dutch, buysen Booze, v. n. == drink. Rel. Ant. ii. p. 175 Bord, sb. == border. Alys. 1270 Borough, sb. [boru]. RG. 72
- 67. Borow, v. a. == defend. Wright’s L. P. pp. 24, 25. part. ‘iborȝe,’ O. and N. 881 Borow, sb. == surety. RG. 472, 497 Borrow, v. a. RG. 393 Borstax, sb. == pick-axe. Pol. S. 151 Bosk, sb. == wood. RG. 547. Fr. bos, bosche Boss, sb. == an ornament of dress. Pol. S. 154. Fr. bosse Bote, sb. See Boot Botemay, sb. == bitumen. Alys. 4763 Botfork, sb. == a crooked stick. Wright’s L. P. p. 110 Both, adj. RG. 376, 445. ‘both two.’ Body and Soul, 120. [bo]. Wright’s L. P. p. 58 Both, == are. See Be In O. and N. 630, 633, the meaning of ‘both’ is uncertain; perhaps a mistake for ‘doth’ Botheler, sb. == peasant, shepherd. Body and Soul, 144; from ‘booth’? Boting, sb. == recompense. Alys. 5711 Bough, sb. [bowe]. RG. 283. [boye], O. and N. 15 Bouk, sb. == body. Alys. 3946. [buc], O. and N. 1130. AS. búce. Germ. bauch Bouked, adj. == protuberant. Alys. 6265 Boulder, sb. == a large stone. HD. 1790
- 68. Boun, adj. == ready. Wright’s L. P. p. 100. Ritson’s AS. viii. 149. ON. búinn. Bound, sb. == boundary. Alys. 5593 Bouning, == making ready. Wright’s L. P. p. 25 Bout, sb. == apparently some female ornament for the face. Pol. S. 154 Bow, sb. RG. 377, 541 Bow, v. a. == bend. pret. ‘buyede.’ RG. 475. ‘beh.’ Wright’s L. P. p. 54. ‘bed,’ 2127 B. —— v. n. == bow or bend. Wright’s L. P. p. 70. AS. búgan. Bowels, sb. Pol. S. 213. Alys. 4668. For the etymology of this word, see Phil. Soc. Trans. for 1856, p. 36 Bower, sb. HD. 2072. Wright’s L. P. p. 114. AS. búr. Bowermaiden, sb. == Rel. Ant. ii. p. 175 Bowiar, sb. == bow-maker. RG. 541 Bowl, sb. K. Horn, 1155 Bowman, sb. RG. 378 Bowshot, sb. Alys. 3491 Box, sb. RG. 456 Boy, sb. Pol. S. 237 Boy, == man. HD. 1899 Brag, adj. == boastful, bold. Wright’s L. P. p. 24 Braid, vb. The following analysis of this difficult verb is taken from Egilson’s Lex. Poet. Septent. s. v. bregða. All the senses here given
- 69. are found in the O. Norse, while the AS. ‘bredan’ apparently is only used in those marked with an asterisk. * I. act. to weave, part. ‘broiden.’ O. and N. 645 II. act. to move a thing from its place. Hence, α. to draw out, as a sword. HD. 1825. part. ‘ybrad’ == drawn, caught. Wright’s L. P. p. 39 β. to brandish, as a sword or spear. Alys. 7373 γ. to pull down. RG. 22. [breide], Alys. 5856 * δ. to seize, or perhaps tear. Rel. S. v. 200. [brede] III. neut. to change, as— α. to awake out of sleep. HD. 1282 β. of any violent motion of body, as to leap. Body and Soul, 46 Braid, sb. == 1. a quick motion, from III. β.; hence, ‘at a breid’ == in an instant. Body and Soul, 182. ON. bragð. 2. a violent struggle or wrench. RG. 22 Brain, sb. RG. 49, 446 Branch, sb. RG. 152 Brand, sb. == a burning mass. Body and Soul, 208 —— == torch. Alys. 5295. [brond]. AS. brand —— == fire. Alys. 1856. [wilde bround] Brased, adj. == of brass. Ps. cvi. 16 Brass, sb. RG. 2, 251
- 70. Bray, sb. == noise. Alys. 2175 Breach, sb. [bruch]. Wright’s L. P. p. 30 Bread, sb. RG. 238 Breadth, sb. [brede]. RG. 385 Break, v. a. 47 B. part. ‘i-broke’ 1005 B. —— v. n. pret. ‘brake.’ 2154 B. —— == to break out (of flesh). 2421 B. Breaking, sb. == breach, gap. Ps. cv. 23 Breast, sb. RG. 419 Breath, sb. Fragm. Sci. 203 Breathe, v. n. Fragm. Sci. 202 Breche, == beech? q. v. Breech, sb. == rump. RG. 322 —— == breeches. 260 B. Breed, v. a. (of a bird). 2 s. pres. ‘breist.’ O. and N. 1631. RG. 177. part. ‘ibred’ == brought up, educated. O. and N. 1722. Body and Soul, 81 Breed, v. n. == spring forth. Wright’s L. P. p. 45 Breist, == breedest. See Breed Breme, adj. == glorious, renowned. Wright’s L. P. pp. 52, 32. AS. breme —— == eager, lustful. O. and N. 202 Brenne, sb. == burning. HD. 1239
- 71. Breth, sb. == wrath. Ps. ii. 5; vi. 2. ON. brædi == anger Breven, v. a. == write down. Pol. S. 156. Brew, v. a. [browe]. RG. 26 Brewer, sb. Rel. S. vii. 35 Brewster, sb. Rel. Ant. ii. p. 176 Breȝe, sb. See Brow. Breze, sb. == gadfly. Ps. civ. 34. AS. brimse Briar, sb. RG. 331 Bridal, sb. Alys. 1071. K. Horn, 1064 Bride, sb. HD. 2131 Bridegroom, sb. [bridegome]. Ps. xviii. 6 Bride, sb. == bridle. Alys. 7626 Bridge, sb. RG. 399 Bridle, sb. RG. 396 Bright, adj. HD. 2131. Wright’s L. P. p. 33 Brim, sb. == brink. 476 β Brimstone, sb. Body and Soul, 219 Bring, v. a. RG. 379. pret. ‘brought.’ RG. 309. part. ‘ybroȝt,’ ‘ibrouȝt.’ RG. 376, 491 Brinie, sb. == cuirass. HD. 1775. Fr. brugne, brugnie. The root is ‘brun’ from ‘brinnan,’ to burn or shine; Cf. OHG. brunna Brink, sb. Alys. 3491. K. Horn, 147 Brise, v. a. == bruise. HD. 1835
- 72. Bristle, sb. Alys. 6621 Bristled, adj. == having bristles. Alys. 5722 Britheling, == worthless, a rascal. Rel. S. vi. 11. Cf. O. Eng. ‘brothell’ Brittene, == cut in pieces? HD. 2700. Cf. ‘brittned,’ in Gloss. to Ormulum. AS. bryttian Broach, sb. (an ornament). RG. 489. Alys. 6842 Broad, adj. RG. 1. [brede], O. and N. 963 —— v. a. == make broad, part. ‘ibroded.’ O. and N. 1310 Broerh, adj. == brittle? Wright’s L. P. p. 23 Brood, sb. RG. 70 Broodful, adj. Ps. cxliii. 13 Brook, sb. RG. 80 Broom, sb. (genista). Alys. 2492 Brost, sb. O. and N. 976, a mistake for ‘prost,’ i.e. ‘priest.’ The Jesus Coll. MS. reads ‘preost’ Broth, sb. RG. 528 Brother, sb. RG. 371, 478 Brouke, v. a. == use, enjoy. HD. 311 AS. brúcan. Germ. brauchen Brow, sb. Wright’s L. P. p. 28. [breȝe]. Ib. p. 34 Brown, adj. RG. 429 —— v. n. == become brown. Alys. 3293 Brun, sb. == a brown jar. K. Horn, 1134
- 73. Brune, sb. == a burning. O. and N. 1153 Brust, adj. == rough, brusque. Pol. S. 151 Brut, adj. == rough? RG. 536 —— == bright. Body and Soul, 57 Bruthen, adj. == fierce, fiercely boiling, ‘a bruthen led.’ Rel. S. v. 242. Connected with ‘breth,’ and AS. brédan, to warm Bu, sb. == buffalo. Alys. 5957 Bu, vb. See Buy Buck, sb. Ritson’s AS. iii. 8 —— == he-goat. Ps. xlix. 13 Buckle, sb. Wright’s L. P. p. 35 Buckler, sb. Alys. 1190 Budel, sb. == messenger. O. and N. 1167. Wright’s L. P. p. 22. AS. bydel Bugging, sb. == a building, or lodging. Pol. S. 151. AS. byggan. ON. byggja Bugle, sb. == buffalo. Alys. 5112 Buglehorn, sb. Alys. 5282 Build, v. a. RG. 439 Bulge, sb. == a lump, hump. Body and Soul, 185 Bulies, == bellows, q. v. Bull, sb. (animal). RG. 116 Bull, (Pope’s bull). RG. 473, 494
- 74. Bullock, sb. Ritson’s AS. iii. 8 Bunting, sb. (the bird). Wright’s L. P. p. 40 Burde, sb. == beard. Alys. 1164 Burdon, sb. == a pilgrim’s staff. K. Horn, 1093. Fr. bourdon Burel, sb. == sackcloth. Alys. 5475. Pol. S. 221. Fr. bure, burel. See Roq. Burgess, sb. RG. 540, 541 Burial, sb. See Buryel Burn, v. a. pret. ‘barnde.’ RG. 380, 511. ‘brende.’ RG. 536. part. pres. ‘berninde.’ RG. 534 Burn, sb. == rivulet. O. and N. 916. AS. byrnan, to burn. Cf. Lat. torrens, from torreo Burst, v. n. pret. ‘barst.’ RG. 437 Burst, sb. == injury. Wright’s L. P. p. 24. AS. byrst Burthen, sb. HD. 807 Bury, v. a. RG. 123. part. ‘y-bured.’ RG. 382. AS. byrgan Burying, sb. RG. 382. [beryng]. Alys. 4624 Buryels, sb. == a tomb, grave. RG. 204. AS. byrgels Busemere, == blasphemy. See Bismere Busily, adv. Ps. cxlii. 7 Busk, v. a. == array. Pol. S. 239 Busy, adj. Alys. 3906 But, adv. 43 B.
- 75. But, prep. == except [bote]. RG. 382. [butent]. Rel. S. ii. 25 —— == without [bute]. O. and N. 184. AS. bútan But, sb. == a put, i.e. cast or throw, HD. 1040 But, part. == contended. HD. 1916. Fr. bouter Butcher, sb. Pol. S. 192 Bute, prep. See But Butler, sb. RG. 187, 438 Butte, sb. == a fish, probably a turbot. HD. 759. The Prompt. Parv. translates it by ‘pecten;’ the Pictorial Vocab., published by Mr Wright, p. 254, has ‘hic turbo’ == ‘a but.’ See N. and Q. 2d S. vi. 382. Sw. butta Butter, sb. HD. 643 Button, sb. Pol. S. 239 —— v. n. == break out. St Swithin, 151. Fr. boutonner. Cotgr. Buxomness, sb. == obedience. RG. 234, 318. AS. buhsomnes, from ‘bugan,’ to bow Buy, v. a. [biggen]. Moral Ode, st. 33. [buggen]. O. and N. 1366. pret. ‘bouȝte.’ RG. 379, 496. ‘bu,’ imper. RG. 390 —— == to exact atonement for. K. Horn, 912 —— == redeem. Ps. xxv. 11 Buyer, sb. == redeemer. Ps. xviii. 15 Buzzard, sb. Alys. 3049 By, prep. == beside (of place). 1213 B. ‘Nolde God that ich bi thé sete’
- 76. —— == according to. 169 B. ‘bi his rede.’ —— == during (of time). 649 B. ‘bi myn ancestors daye.’ 2498 B. ‘bi a Tuesdai’ —— == against. 871 B. ‘bi the Bischop of L. thulke word he sede.’ Cf. 1 Cor. iv. 4 —— == concerning, of. O. and N. 46 By. For verbs compounded with ‘By,’ see under ‘Bi’ Bycase, adv. == by chance. RG. 490 Byefþe. See Behoof Byquide. See Bequest Byȝyte. See Beget
- 77. C. Cable, sb. RG. 148 Cacherel, sb. == catch poll. Pol. S. 151 Cage, sb. Alys. 5011 Caitiff, sb. Body and Soul, 229 Cake, sb. Cok. 55 Cales, sb. == a kind of serpent. Alys. 7094 Calf, sb. (the animal). Alys. 6351 Call, v. n. Wright’s L. P. p. 59 Call, sb. == cap worn on the head. Pol. S. 158. Fr. cale Caluȝ, adj. == bald. Alys. 5950. AS. calo, caluw Camel, sb. Alys. 854 Can, vb. == am able [con]. Wright’s L. P. p. 82. [cunne]. 2 s. pres. ‘cost.’ Wright’s L. P. p. 91. O. and N. 47. pret. ‘cowþe.’ RG. 29 —— == know [con]. RG. 443. [cunne]. O. and N. 48. 2 s. pres. ‘canst.’ O. and N. 560 Candle, sb. RG. 290, 561 Candlemas, sb. St Dunstan, 3 Canel, sb. == cinnamon. Wright’s L. P. p. 27. Fr canelle. Lat. canna Cankerfret, adj. RG. 299 Canon, sb. RG. 510
- 78. Capel, sb. == horse, nag. Cok. 32. Lat. caballus Capelclawer, sb. == horse-scrubber. Pol. S. 239 Capital, sb. (of a column). Cok. 67 Carbuncle, sb. Alys. 5252. HD. 2145 Cardinal, sb. 1280 B. Care, sb. RG. 457 Care, v. n. == be anxious. RG. 71. Wright’s L. P. p. 54 Careful, adj. == full of care. 639 B. Carie, sb. == carat. Alys. 6695 Carke, v. n. == pine away. Wright’s L. P. p. 54 Carol, sb. RG. 53 Carol, v. n. Alys. 196, 1045 Caronye, sb. == carcass. RG. 265 Carp, v. n. == complain. Pol. S. 149 Carpenter, sb. RG. 537 Carrion, sb. (caraing). Pol. S. 203 Cart, sb. RG. 189 Cartload, sb. HD. 895 Cartstave, sb. RG. 99 Carve, v. a. RG. 560. == cut, flay. part. ‘corven.’ Wright’s L. P. p. 35. ‘curven.’ HD. 189 Case, sb. == chance, event. RG. 528
- 79. —— == condition. Alys. 4428 Cast, v. a. RG. 511, 375 Castle, sb. RG. 371, 510; pl. ‘kasteles’ == tents. Ps. lxxvii. 28 Cat, sb. Alys. 5275 Catathleba (κατώβλεπας), == a noxious monster, mentioned in Alys. 6564. See Pliny, H. N. viii. 32 Catch, v. a. RG. 28. pret. ‘caught.’ RG. 375. part. ‘cacchynge.’ RG. 265 Cathedral, adj. RG. 282 Caudle, sb. RG. 561 Cauldron, sb. 158 β Caution, sb. == surety. RG. 506 —— == quarter in battle. Alys. 2811 Cavenard, sb. == villain. HD. 2389. The form ‘caynard’ is found in Wright’s L. P. p. 110. Fr. caignard. Cotgr. Cayre, v. a. == turn. part. ‘ycayred.’ Wright’s L. P. p. 37. AS. cerran. Germ. kehren Caynard. See Cavenard Cayser, sb. == emperor. HD. 1317. Wright’s L. P. p. 32 Cayvar, adj. == hollow? Alys. 6062 Cedar, sb. Ps. ciii. 16 Cel, sb. == seal. RG. 77 Celadoyne, sb. See Celandine
- 80. Celandine, sb. == the flower. Wright’s L. P. p. 26. Lat. chelidonium. It is the ‘ranunculus ficaria’ of botanists Cell, sb. RG. 233 Cellar, sb. 287 B. Cement, sb. Alys. 6177 Censer, sb. Marg. 75 Cerge, sb. == a taper. HD. 594. ON. kérti. Germ. kerze Cert, adv. == certainly. Alys. 5803 Certain, adj. == fixed, ascertained. RG. 378, 552 Certés, adv. 898 B. Cestred, == lodged, concealed. Ps. lxxiii. 20; cxxxviii. 12. AS. ceaster Chaffare, sb. == merchandise. RG. 539. AS. ceápian Chair, sb. RG. 321 Chaisel, sb. == a woman’s upper garment. Alys. 279. ‘espéce de vétement.’ Roq. s. v. SS. cheisil. Fr. cheinsil, v. Roq. s. v. chainse, and Gloss. Rem. to Laȝ. iii. 502 Chalandre, sb. == goldfinch. Cok. 95 Chalcedony, sb. Cok. 92 Chalen, sb. == chill, cold. Alys. 4834 Chalice, sb. RG. 489. HD. 187 Chalktrap, sb. == pit or snare. Alys. 6070 Challenge, v. a. RG. 279, 451 Chamber, sb. 452 B.
- 81. Chamberlain. RG. 390, 490 Champion, sb. HD. 1015 Chance, sb. == condition, fortune. RG. 465 —— == chance [cheance]. RG. 210 Chancellor, sb. RG. 540, 468 Chancellory, sb. == office of chancellor. 452 B. Chane, vb. pret. == cleft. Alys. 2228. AS. cínan. perf. cán. The ‘ch’ appears in ‘tochan,’ the pret. of ‘tocínan,’ in Laȝamon, ii. 468. Weber wrongly derives the word from Fr. choir, and makes it mean ‘fell’ Change, sb. RG. 493 Change, vb. a. RG. 548 Chantment, sb. == enchantment. RG. 28, 149 Chapel, sb. RG. 472, 473 Chapitle, sb. == chapter of a cathedral RG. 473 Chaplain, sb. 961 B. Chapman, sb. RG. 539 Chapter, sb. (of a cathedral). 601 B. Char, sb. == turn, movement. Body and Soul, 79. Hence ‘ȝeynchar’ == repentance. Wright’s L. P. p. 46. SS. charren. AS. cérran, cérre. Germ. kehren Charge, v. a. == load. RG. 13. part. ‘icharged.’ Pol. S. 195 —— sb. == load, weight. RG. 416 —— == expense. RG. 189 Charity, sb. Pol. S. 202. ‘par charité.’ 1811 B.
- 82. Charm, sb. == spell. Alys. 81 Charming, sb. == spell. Alys. 404 Charreye, sb. == car. Alys. 5097 Charter, sb. RG. 477, 498 Chase, sb. == hunting. RG. 6 Chaste, adj. 154 B.; [cheste]. Alys. 7050. ‘chaster.’ RG. 191 Chaste, v. a. == chastise. RG. 134 Chastise, v. a. RG. 420 Chasuble, sb. == priest’s robe. 953 B. Fr. casule. Ital. casupola Chasur, sb. == horse for hunting. Signa ante Jud. 110. Fr. chaceor Chaterestre, sb. == a female chatterer. O. and N. 655 Chattels, sb. [chateus]. RG. 471, 569. Another form of ‘cattle’ Chattering, sb. O. and N. 744 Chaumpebataile, sb. == battle-field. Alys. 5553 Chavling, sb. == jawing. O. and N. 284, 296 Chawl, sb. == jaw. Body and Soul, 189. Pol. S. 154. AS. ceafl. SS. chevele. pl. chæfles Chawl, v. n. == to chide, jaw. Pol. S. 240 Cheap, sb. == haggling? Wright’s L. P. p. 39 Cheap, v. a. == buy. Pol. S. 159. AS. ceápian Cheaping, sb. == market. Pol. S. 151 Cheek, sb. Wright’s L. P. p. 34
- 83. Cheer, sb. == comfort. 473 B. —— == countenance. RG. 332 Cheese, sb. HD. 643 Chelde, sb. == chill, cold. Alys. 5501 Cheole, sb. == hair. M. Ode, 182. Fr. chevol Chepe, Cheping. See Cheap, Cheaping Chequer, sb. == chess. RG. 192; or perhaps ‘the chessboard’ Cherde, vb. pret. == turned, came. O. and N. 1656. AS. cyrran, cérran Chere, adj. == high? ‘the chere men of the land.’ RG. 166. Cheson, sb. == occasion. Alys. 3930 Chess, sb. Alys. 2096 Chest, sb. == coffin. RG. 50 Cheste, sb. == strife. Alys. 29. AS. ceást Chete, == a chewet, or pie. Wright’s L. P. p. 31 Cheui, an error for ‘cheve.’ RG. 94 Cheve, v. n. == succeed in a thing. 856 B. Fr. chevir Chide, v. a. 2 s. pres. ‘chist.’ O. and N. 1329. AS. cídan —— v. n. RG. 390 —— == dispute. O. and N. 287 Chief, sb. == chieftain. 1003 B. Chief, adj. == ‘to hold in chief,’ a law term, applied to those tenants who held their fiefs direct from the king; ‘tenants in capite.’ RG. 472
- 84. —— == principal. St Swithin, 22 Chieftain, sb. [cheventeyn]. RG. 386, 400 Chilce, sb. == childishness? M. Ode, 4. Formed from ‘child,’ as ‘milce’ from ‘mild’ Child, sb. RG. 392, 441; [chil]. O. and N. 1438 Child, v. n. == bring forth a child. Alys. 604 Childbed, sb. RG. 379 Childering, sb. == bringing forth a child. Rel. S. ii. 7 Chill, sb. RG. 7 Chimbe, sb. == cymbal. Ps. cl. 5 Chime, sb. (of bells). Alys. 1852. Dan. kime Chin, sb. 522 β Chinche, adj. == niggardly. HD. 1763. Fr. chice == avarice Chirchegong, == churchgoing. RG. 380. Cf. ‘idelgong’ Chirm, sb. == chirping and screaming of birds. O. and N. 305. AS. cyrm Chirurgeon, sb. RG. 566 Chivalry, sb. == prowess. RG. 413 Chivauché, sb. == an expedition, a body of men. Ritson’s AS. viii. 141. Fr. chevauchée, from cheval. Choice, sb. RG. 111 Chokering, sb. == a low chattering. O. and N. 504 Cholle, == shall. RG. 379, in the compound form ‘ycholle’
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