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STOCHASTIC GEOMETRY:
Selected Topics

This page intentionally left blank

STOCHASTIC GEOMETRY:
Selected Topics
by
Viktor Beneš
Faculty of Mathematics and Physics
Charles University, Prague
Jan Rataj
Faculty of Mathematics and Physics
Charles University, Prague
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: 1-4020-8103-0
Print ISBN: 1-4020-8102-2
Print ©2004 Kluwer Academic Publishers
All rights reserved
No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,
mechanical, recording, or otherwise, without written consent from the Publisher
Created in the United States of America
Boston
©2004 Springer Science + Business Media, Inc.
Visit Springer's eBookstore at: http://www.ebooks.kluweronline.com
and the Springer Global Website Online at: http://www.springeronline.com

Contents
Preface
Acknowledgments
1. PRELIMINARIES
1.1 Geometry and measure in the Euclidean space
1.1.1
1.1.2
1.1.3
1.1.4
Measures
Convex bodies
Hausdorff measures and rectifiable sets
Integral geometry
1.2 Probability and statistics
1.2.1
1.2.2
1.2.3
Markov chains
Markov chain Monte Carlo
Point estimation
2. RANDOM MEASURES AND POINT PROCESSES
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Basic definitions
Palm distributions
Poisson process
Finite point processes
Stationary random measures on
Application of point processes in epidemiology
Weighted random measures, marked point processes
Stationary processes of particles
Flat processes
ix
xi
1
1
2
3
5
8
12
14
16
17
21
22
25
28
30
32
35
38
40
43

vi STOCHASTIC GEOMETRY
3. RANDOM FIBRE AND SURFACE SYSTEMS
3.1 Geometric models
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
3.1.6
Projection integral-geometric measures
The Campbell measure and first order properties
Second-order properties
and Palm distributions
Poisson process
Flat processes
3.2 Intensity estimators
3.2.1
3.2.2
3.2.3
Direct probes
Indirect probes
Application - fibre systems in soil
3.3 Projection measure estimation
3.3.1
3.3.2
Convergence in quadratic mean
Examples
3.4 Best unbiased estimators of intensity
3.4.1
3.4.2
3.4.3
3.4.4
Poisson line processes
Poisson particle processes
Comparison of estimators of length intensity of
Poisson segment processes
Asymptotic normality
4. VERTICAL SAMPLING SCHEMES
4.1 Randomized sampling
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
4.1.6
IUR sampling
Application - effect of steel radiation
VUR sampling
Variances of estimation of length
Variances of estimation of surface area
Cycloidal probes
4.2 Design-based approach
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
VUR sampling design
Further properties of intensity estimators
Estimation of average particle size
Estimation of integral mixed surface curvature
Gradient structures
Microstructure of enamel coatings
45
47
48
50
52
55
58
60
61
63
67
72
75
76
80
81
82
85
86
88
93
95
95
97
99
102
104
111
114
114
117
120
124
130
132

Contents vii
5. FIBRE AND SURFACE ANISOTROPY
5.1
5.2
Introduction
Analytical approach
5.2.1
5.2.2
5.2.3
Intersection with in
Relating roses of directions and intersections
Estimation of the rose of directions
5.3 Convex geometry approach
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
Steiner compact in
Poisson line process.
Curved test systems
Steiner compact in
Anisotropy estimation using MCMC
5.4 Orientation-dependent direction distribution
6. PARTICLE SYSTEMS
6.1 Stereological unfolding
6.1.1
6.1.2
6.1.3
Planar sections of a single particle
Planar sections of stationary particle processes
Unfolding of particle parameters
6.2 Bivariate unfolding
6.2.1
6.2.2
6.2.3
Platelike particles
Numerical solution
Analysis of microcracks in materials
6.3 Trivariate unfolding
6.3.1
6.3.2
6.3.3
6.3.4
Oblate spheroids
Prolate spheroids
Trivariate unfolding, EM algorithm
Damage initiation in aluminium alloys
6.4 Stereology of extremes
6.4.1
6.4.2
6.4.3
6.4.4
6.4.5
6.4.6
6.4.7
Sample extremes – domain of attraction
Normalizing constants
Extremal size in the corpuscule problem
Shape factor of spheroidal particles
Prediction of extremal shape factor
Farlie-Gumbel-Morgenstern distribution
Simulation study of shape factor extremes
135
135
136
136
138
140
143
145
150
152
155
159
161
169
169
170
171
173
176
176
179
181
182
184
188
191
193
196
197
198
199
200
203
205
207

viii STOCHASTIC GEOMETRY
References
Index
211
219

Preface
Since the seventies years of the past century, stimulated namely by the
ingenious collection edited by Harding and Kendall [47] and Matheron’s
monograph [69], stochastic geometry is a field of rapidly increasing in-
terest. Based on the current achievements of geometry, probability and
measure theory, it enables modeling of two- and three-dimensional ran-
dom objects with interactions as they appear in microstructure of mate-
rials, biological tissues, macroscopically in soil, geological sediments, etc.
In combination with spatial statistics it is used for the solution of prac-
tical problems such as description of spatial arrangement and estimation
of object characteristics. A related field is stereology which makes infer-
ence on the structures based on lower-dimensional observations.
The subject of stochastic geometry and stereology is nowadays broadly
developed so that it can be hardly covered by a single monograph. This
was successfuly tried in the eighties by Stoyan et al. [109] (the first
edition appeared in 1987), recently, however, specialized books appear
more frequently, as those by Schneider & Weil [103], Vedel-Jensen [116],
Howard & Reed [54], Van Lieshout [115], Barndorff-Nielsen et al. [2],
Ohser & Mücklich [86] and Møller & Waagepetersen [80]. This list might
be followed by volumes on the shape theory and random tessellations.
We tried to continue this series by collecting several recently studied
topics of stochastic geometry and stereology, with accents on fibre and
surface systems, particle systems, estimation of intensities, anisotropy
analysis and statistics of particle characteristics.
Like in all applied areas, a close cooperation between theoretical math-
ematicians working in stochastic geometry and related fields and scien-
tists doing applied research is necessary, and there are several activities
aiming at a satisfaction of this need. Among them, we may mention
regular stereological congresses organized by International Society for
Stereology, where biologists, medical doctors, material scientists and

mathematicians meet together, further the joint workshops of mathe-
matitians and physicists interested in stochastic geometry organized by
D. Stoyan (see [74]).
The present book is an attempt to present the theory in a concise
mathematical way, but illustrated with a number of practical demon-
strations on simulated or real data.
After the first chapter presenting necessary background from measure
theory, convex geometry, probability and statistics, Chapter 2 is devoted
to an overview of the basic notions and results on random sets, random
measures and point processes in the euclidean space (we refer here fre-
quently to the monograph of Daley & Vere-Jones [23]). Section 3 deals
with stationary random fibre and surface systems and the estimation of
their intensities. In our terminology, a random fibre (surface) system is
a random closed set, whereas the notion of a fibre (surface) process is
reserved for genuine processes of fibres (surfaces). Using an approach of
geometric measure theory, fibres (surfaces) are modelled by Hausdorff
rectifiable sets, as suggested by Zähle [125] already in 1982, but not
widely accepted in stochastic geometry so far.
Chapter 4 is devoted to an important method of geometric sampling
called “vertical” and originated in the eighties by Baddeley, Cruz-Orive
and Gundersen (see [1, 4]). Vertical sampling is a promising alternative
to isotropic uniform random (IUR) sampling which is hardly applicable
to real structures. Vertical sampling designs are applied to the intensity
estimation of random fibre and surface systems and following the joint
research with Gokhale [39], [51] to the estimation of some other char-
acteristics as particle mean width or integral mixed curvature. We use
both model- and design-based approaches in this chapter.
The analysis of anisotropy of a random fibre or surface system is the
contents of Chapter 5. We focus on both planar and spatial fibre sys-
tems and spatial surface systems. An overview is presented of different
methods solving the inversion of the well-known formula connecting the
rose of intersections with the rose of directions (see (5.10)). The esti-
mation of the orientation-dependent rose of normal directions (for the
boundary of a full-dimensional body in the space) is considered as well.
Section 6 deals with the stereology of particle systems, reviewing and
developing some classical methods of unfolding of particle parameters
using data from planar sections. Including particle orientations among
parameters requires again vertical sections to be employed for sampling.
Finally, applying the statistical theory of extremes it is shown how to
detect extremal characteristics of particles.
x STOCHASTIC GEOMETRY

Acknowledgments
The research work included was supported by several grants, most
recently by the Czech Ministery of Education project MSM 113200008,
the Grant Agency of the Czech Republic, project 201/03/0946, Grant
Agency of the Academy of Sciences of the Czech Republic, project IAA
1057201 and Grant Agency of Charles University, project No. 283/2003
/B-MAT/MFF. PhD students and post-doctorands from the Faculty of
Mathematics and Physics, Charles University in Prague, participated on
the solution of grants and parts of their papers are reported. This con-
cerns D. Hlubinka, M. Hlawiczková, K. Bodlák,
Z. Pawlas and M. Prokešová. Particular thanks are addressed to
Pawlas and Josef Machek for corrections and comments to the book.
Pawlas moreover edited most of the figures.

Chapter 1
PRELIMINARIES
1.1. Geometry and measure in the Euclidean
space
Let denote the Euclidean space with Euclidean norm
and scalar product By we shall denote the closed ball
the Minkowski sum of A and B and we write shortly instead of
(the translate of A by the vector Further,
denotes the Minkowski subtraction of A and B. By
with centre and radius and we shall write briefly The
symbol denotes the unit sphere,
the group of rotations in and the Grassmannian of
linear subspaces of The space is equipped with
the unique rotation invariant (uniform) probability distribution denoted
by U.
For two subsets A, B of we denote by
we
denote the central reflection of B and the set is called the
dilation, erosion of A with B, respectively. It follows from the definition
that if and only if the translate hits A, whereas
if and only if is contained in A.
The symbol denotes the segment with end-points

2 STOCHASTIC GEOMETRY
1.1.1 Measures
Under a measure we always understand a nonnegative and
set functional. A Borel measure in is a measure defined on the
of Borel subsets of The symbol denotes the Dirac
measure concentrated in i.e., where is the
characteristic function of the set B (i.e., for and
otherwise).
If is a measure on a measurable space (A, and
a measurable mapping, then denotes the of i.e.,
For two measures on (A, we say that is absolutely continuous
with respect to (written as if any set is also a
set. If this holds for measures then there exists a
everywhere unique measurable function (called a density of
with respect to such that
The support of a measure on is defined as
supp is the smallest closed set such that vanishes on its complement.
The convolution oftwo measures on is the Borel measure
in
where is the product measure in and
is the usual operation af addition. Using (1.1) and the Fubini theorem,
we get the standard formula
The definition of the convolution can be applied analogously for any two
measures on a measurable space equipped with the addition operation
which should be measurable.
By we shall always denote the Lebesgue measure in and
denotes the volume of the unit ball in For the integration
with respect to the Lebesgue measure we shall sometimes write only
instead of
a measure on (E, given by

Preliminaries 3
1.1.2 Convex bodies
Let be the system of all compact convex sets, nonempty compact
convex sets in respectively. A set is called a convex body. If
then for each there is exactly one number such
that the hyperplane (line in plane in
intersects K and for each This hyperplane is
called the support hyperplane and the function is the
support function (restricted to of K. Equivalently, one can define
Its geometrical meaning is the signed distance of the support hyperplane
from the origin of coordinates. The sum
is the width of K - the distance between the parallel support
hyperplanes, see Fig. 1.1. An important property of is its ad-
ditivity in the first argument:
convex body K is centrally symmetric if for some translate
of K, i.e., if K has a centre of symmetry. In what follows, mostly convex
bodies that possess a centre of symmetry will be considered.
The Minkowski sum of finitely many centred line segments is called
a zonotope. Besides of being centrally symmetric, in also its two-
dimensional faces are centrally symmetric. Consequently, regular octa-
hedron, icosahedron and pentagonal dodecahedron are not zonotopes.

On the other hand, in all centrally symmetric polygons are zono-
topes.
Consider a zonotope
where Its support function is given by
and, conversely, a body with the support function (1.6) is a
zonotope with the centre in the origin.
We shall use the standard notation for the space of all nonempty
compact subsets of equipped with the Hausdorff metric (see [69, 43])
(dist is the distance of a point from the set L).
The corresponding convergence is denoted as A set
is called a zonoid if it is a of a sequence of zonotopes.
A convex body Z is a zonoid if and only if its support function has a
representation
for an even measure on (see [41, Theorem 2.1]). The measure
is called the generating measure of Z and it is unique as shown in
[69, Theorem 4.5.1], see also [41]. For the zonotope (1.5) we have the
generating measure
where
Zonotopes and zonoids have several interesting properties and wide
applications (see [41], [101]), e.g. the polytopes filling (tiling) by
translations are obligatory zonotopes (cubes, rhombic dodecahedrons,
tetrakaidecahedrons).
EXERCISE 1.1 Express the support function of a line segment in in
polar coordinates.

Preliminaries 5
EXERCISE 1.2 Verify the additivity formula
EXERCISE 1.3 Verify the following formula for the Hausdorff distance
of two convex bodies:
EXERCISE 1.4 Show that the family of convex bodies is closed in
with respect to the Hausdorff metric.
1.1.3 Hausdorff measures and rectifiable sets
In this subsection we give a survey of some notions and results from
geometric measure theory which can be found in Federer [31] or Mat-
tila [70]. An instructive treatment of the area and coarea formulae for
smooth sets with applications in stochastic geometry was presented by
Vedel-Jensen [116].
Let be fixed. The Hausdorff measure of order
in is defined as
where diam denotes the diameter of and the infimum is taken over
all at most countable coverings of A with (any) sets of diameters less
or equal to Equation (1.10) may be applied to any subset A of
defining as an outer measure (called a measure in [31, §2.1.2]). The
outer measure becomes a measure when restricted to the family of
sets which encompass the family of Borel sets. It can
be shown that is Borel regular (i.e., for any there exists a
Borel set with motion invariant, homogeneous
of order and, in particular, is the counting measure and
Note that extends the standard differential-geometric
measure defined on smooth submanifolds of see
e.g. [97].
We call a subset if it is a Lipschitz image of a
bounded subset of (a mapping is Lipschitz if there exists a constant
M such that
if
A set is
1) A is
2)
for any from the domain of

3) with and for
Finally, is if is for any
compact.
Any piecewise manifold in is
fiable, but the class of sets is substantially larger. For
example, the distance function from any subset of (even a fractal
one) is Lipschitz and, hence, its graph is even 1-rectifiable. For the
purposes of stochastic geometry, sets have the important
property that the rectifiability is inherited by sections almost surely (cf.
Theorems 1.11, 1.12).
Unlike the case of (piecewise) smooth manifolds, sets
need not possess tangent planes in the usual sense at
almost all points. Nevertheless, this property is true with a suitably
adjusted definition of tangent vectors.
The tangent cone of a set at is the closed cone in
defined by the following property: and
a vector belongs to if and only if for any there
exists with and
The tangent cone of A at is then given by
where
is the (upper) density of E in The set is
again a closed cone in which is in general a subset of
Roughly speaking, we can say that neglects the “lower than
components” of A, see Fig. 1.2. If A is a
coincides with the classical tan-
gent at The importance of the approximate tangent cones
follows from the following theorem.
THEOREM 1.5 ([31, §3.2.19]) If A is then for
all is a subspace of
Let be and let be Lipschitz. It
is not difficult to show that is (in as well. It

Preliminaries 7
can further be shown that can be approximated by a
mapping on such that for all
If is such a point and if, furthermore, is a
subspace, we define the approximate differential of at
ap as the restriction of thedifferential to the approximate
tangent subspace (the correctness can be shown). For
the approximate Jacobian of at is then defined as
where the supremum is taken over all unit C in
Note that, if A is a and is differentiate on A, ap
is the classical differential and ap the classical Jacobian of at
Now we can formulate the general area-coarea formula:
THEOREM 1.6 ([31, §§3.2.20,22]) Let be
and let be a Lips-
chitz mapping and let be a nonnegative measurable function.
Then

EXERCISE 1.7 If
where represents the matrix of partial derivatives of at a.
EXERCISE 1.8 If is the restriction of a differentiable mapping
to a set then ap
for all
EXERCISE 1.9 Show that the boundary of a convex body in is
EXERCISE 1.10 Let be a curve in of finite length and let
denote the orthogonal projection onto asubspace Assume
that the projection is injective everywhere on By using
Theorem 1.6, show that
where
1.1.4 Integral geometry
The object of integral geometry are mainly formulas involving kinema-
tic (translative) integrals of some geometric quantities. As classical ref-
erence, the book of Santaló [97] serves, whereas for our purposes, later
treatment using the measure theoretic language is more appropriate (e.g.
[104, 102]).
One of the simplest integral-geometric formulas follows directly from
the Fubini theorem. If A is a measurable subset of and F a in
(a affine subspace), then
where denotes the subspace perpendicular to F. Including an additi-
onal integration over rotations, one obtains
where is the space of all in and the motion invariant
measure on normed as the product of the uniform probability distri-
bution on with Lebesgue measure (i.e., we can
is differentiable at a and then
is the tangent direction of at

Preliminaries 9
write if is the unique decomposition
of into a linear subspace L and a shift An analogous
formula for the volume of the intersection of two bodies follows again
from the Fubini theorem:
where is the group of all euclidean motions in (i.e., compositions of
rotations and translations) and the invariant measure corresponding
to the product of the rotation invariant probability distribution over
the group of rotations and the Lebesgue measure over
translations.
Kinematic formulas can be written also for the Hausdorff measure of
lower-dimensional (rectifiable) sets. We present here, for illustration, a
result of this type due to Zähle [125, §1.5.1].
THEOREM 1.11 Let be natural numbers and let A be an
and B an subsets of such that their cartesian
product A × B is Then is
for all motions and we have
where (denoting the Euler gamma function)
Translation formulas for the Hausdorff measure are more involved,
including integration over Jacobians. We present a particular version
here which will be used later. To do this, we need some notation. Let
M, N be two linear subspaces of of dimensions respectively,
with and let be orthonormal bases
of M, N such that is a basis of We
shall denote by [M, N] the volume of the parallelepiped
spanned by Note that if then
THEOREM 1.12 Let A, B be as in Theorem 1.11. Then
is for all and we have

The theorem can be proved by applying the coarea formula (Theo-
rem 1.6) to the function defined on A × B (for details, see
[123, 125]).
More important are integral-geometric formulas for “second-order”
(depending on second derivatives) quantities as quermassintegrals, intrin-
sic volumes or curvature measures. In order to define meaningfully these
notions, we have to restrict ourselves to a smaller class of sets, e.g. to
convex or polyconvex bodies, sets with smooth boundaries, or some
generalizations of these. We start for simplicity with convex bodies.
Given a convex body and we define the
intrinsic volume of K by
(here stands for the orthogonal projection to L). After renorming
and reindexing, we get the classical quermassintegrals
(the additional upper index at W indicates the dependence on the dimen-
sion of the embedding space). The intrinsic volumes can be defined also
by means of the
A local version of the Steiner fomula makes it possible to introduce
curvature measures of K as local variants of the intrinsic volumes (see
[99]). If the boundary of K, is we can also express the
intrinsic volumes as integrals of certain functions of principal curvatures.
We shall illustrate this fact only on the example of a smooth convex body
in let denote the principal curvatures of K at
and denote
the Gauss, mean (respectively) curvature of K at Then we have

Preliminaries 11
Formulas (1.16), (1.17) and (1.18) can be applied as definitions of intrin-
sic volumes for setswith smooth boundaries (not necessarily convex).
Note that local curvatures can be defined also for certain nonsmooth
bodies (e.g. convex sets), but the integrals should be performed then
over the unit normal bundle instead of the boundary only (see [126]).
On the other hand, formulas (1.17) and (1.18) together with our def-
inition of intrinsic volumes can be rewritten in the form
which are known as Cauchy (or Kubota) formulas; here we use the clas-
sical notations M(K) for the integral of mean curvature over and S
for the surface area content.
Intrinsic volumes can be extended to polyconvex sets (finite unions of
convex bodies) by additivity (i.e., the property
Even after the extension, the following characteristic
properties remain valid: is the Euler-Poincaré characteristic,
one half of the surface content (in case of a full-dimensional set), and
is the volume (Lebesgue measure). Of course, the Cauchy formulae are
not valid for general polyconvex sets.
The basic integral-geometric relation for intrinsic volumes is
THEOREM 1.13 (PRINCIPAL KINEMATIC FORMULA) Let K, L be polycon-
vex sets in Then for we have
(the constant is defined in Theorem 1.11).
We remark that the principal kinematic formula holds for all reason-
able extensions of intrinsic volumes and also that an appropriate gener-
alization is true for the local versions (curvature measures). Replacing
the second polyconvex set with a flat we obtain
THEOREM 1.14 (CROFTON FORMULA) For a polyconvex set K in
and for with we have

EXERCISE 1.15 Let M, N be linear subspaces of of dimensions
respectively. Then [M, N] can equivalently be defined as the
volume of the parallelepiped spanned by any orthonormal
bases of the complements and
EXERCISE 1.16 Applying Theorem 1.11, show that the translative inte-
gral of the number of intersection points of a curve
with a unit circle in equals
EXERCISE 1.17 Compute the intrinsic volumes of a two- and three-
dimensional ball.
EXERCISE 1.18 Using mathematical induction, show the following iter-
ative version of the principal kinematic formula valid for convex
bodies in
with the constants
1.2. Probability and statistics
In this section, Pr) will denote a (fixed) abstract probability
space, i.e., is a of subsets of and Pr a probability measure
on A measurable mapping X of into a measurable space (T,
is called a random element in T. The distribution of X is a probability
measure on cf. (1.1). Specially, if we call X a random
variable. Standard symbols are used for the expectation of a random
variable variance covariance
cov (X,Y) = E(X – EX)(Y – EY) of two random variables. For any
random vector the distribution function is defined as
A sequence of random elements converges to a random element
X almost surely (a.s.) if

Preliminaries 13
For random elements in a metric space (T, we say that
the sequence converges in probability to X (denoted if
for any
Almost sure convergence implies convergence in probability. A sequence
of random variables with partial sums is said to obey
the strong (weak) law of large numbers if converges almost surely (in
probability) to a constant.
Let be the Borel on a metric space (T, A sequence
of probability measures on converges weakly to a probability
measure (we write if for every bounded
continuous function A sequence of random elements
converges in distribution to a random element X (denoted if
Convergence in probability implies convergence in
distribution and both concepts coincide if X is almost surely a constant.
We shall denote by the Gaussian distribution with mean
and variance Instead of where X is Gaussian we sometimes
write We recall the classical central limt theorem for a
sequence of independent identically distributed (i.i.d.) random variables,
see e.g. [58, Proposition 4.9].
PROPOSITION 1.1 (LÉVY-LINDEBERG) Let be i.i.d. random
variables with and Then
For
A sequence of random variables converges to X in if
Convergence in implies convergence in probability.
The converse implication is not true in general but it holds under an
additional assumption. A system of random variables is
said to be uniformly integrable if
For the proof of the following result, see e.g. [58, Proposition 3.12].
let be the class of random variables X with

PROPOSITION 1.2 Let X, be in and let Then
in if and only if the sequence is uniformly
integrable.
In the following subsection on Markov chains we shall need the notion
of a probability kernel.
DEFINITION 1.19 Let (T, (E, be measurable spaces. A (probabil-
ity) kernel from (T, to (E, is a mapping such
that (i) is measurable for each (ii) is a
(probability) measure for all
1.2.1 Markov chains
The background of Markov chains on arbitrary state spaces is briefly
described. All the notions and statements mentioned in this subsection
can be found in [75].
Let (E, be a Polish space (i.e., separable complete metric space)
with Borel Let be a probability measure on and P a
probability kernel from (E, to (E,
A collection ofrandom elements in E is called a (homoge-
neous) Markov chain with transition kernel P and initial distribution
if for any integer and for any it holds
The power of the kernel P is defined by the recursive formula
where we set The value is interpreted as
the probability that the chain gets from state to A in steps.
The random variable is called the return
time to a set A Markov chain Y is with a probability
measure on (E, if implies for
all According to [75, p. 88], for Y there exists a
maximal (w.r.t. partial ordering probability measure such that Y
is Denote
A set is a small set if there exist and a probability
measure such that for all and it holds

Preliminaries 15
A Markov chain is called aperiodic if for some small set
the greatest common divisor of those for which (1.23)
holds for some is 1. Denote A set is Harris
recurrent if
A Markov chain Y is Harris recurrent if each is
Harris recurrent.
A measure on is invariant (w.r.t. the kernel P) if
for each A Markov chain Y is called positive if it
has an invariant probability measure.
The Markov chain Y with an invariant probability measure is called
ergodic if
for all An aperiodic Harris recurrent Markov chain is ergodic if
and only if it is positive. Further equivalent conditions for ergodicity are
stated in [75, p. 309].
A Markov chain Y with invariant probability measure is called ge-
ometrically ergodic if there exists a finite measurable function M on E
and such that
for any integer and all If, in addition, is bounded, Y is
said to be uniformly ergodic. The chain Y is uniformly ergodic if and
only if E is a small set. Characterizations of geometric ergodicity can
be found in [75].
Let Y be a positive Markov chain and a real measurable function
on E. Denote
and let be the expectation, variance of respectively,
where X is a random element with distribution
A Harris recurrent positive chain Y satisfies the strong law of large
numbers:

A positive Markov chain Y is reversible if for any A, it holds
If Y is geometrically ergodic, reversible and then the central
limit theorem holds:
where is finite and
the initial variable is assumed to have the distribution
1.2.2 Markov chain Monte Carlo
Let (E, be a Polish state space with Borel and a tar-
get probability measure on For the case when it is impossible to
simulate directly from the target distribution we discuss methods of
the construction of an ergodic Markov chain Y with invariant measure
Corresponding simulation techniques are called Markov chain Monte
Carlo (MCMC). In fact, we restrict ourselves to one of them called the
Metropolis-Hastings algorithm, for other methods such as Gibbs sam-
pler, see [34].
Let the target distribution have a density with respect to a ref-
erence measure and denote Let Q be a prob-
ability kernel with density i.e.,
for Define
for otherwise. The algorithm starts in an
arbitrary initial state If the Markov chain state at is
a candidate is simulated from the distribution With
probability the candidate is accepted, otherwise it is rejected and
we set The algorithm almost surely does not leave the
knowledge of up to a multiplicative constant is sufficient.
Define for otherwise. Put
(probability that the chain does not leave in
a single step). Then the transition kernel of the simulated chain is

Preliminaries 17
The detailed balance condition
is fulfilled which implies reversibility, it follows that is an invariant
distribution for the chain Y.
EXAMPLE 1.20 Let be the Lebesgue measure and
a probability density on E. If Z is simulated from the distribution
and then the proposal density is and
Q is a kernel of a random walk. Therefore it is called the Metropolis
random walk algorithm. In the case of symmetry (i.e.,
for all it holds therefore a candidate
with is always accepted.
EXAMPLE 1.21 The Langevin-Hastings variant of the algorithm makes
use of the information from the gradient of the density of the target
distribution. In it is e.g.
1.2.3 Point estimation
In this subsection, some notions from the statistical estimation theory
will be recalled (see [66]).
Let an experiment be given, where is the sample space,
a on and a parametric system of
probability measures on the parameter space being a Polish space.
A random observable X is taking values on according to the distri-
bution a realization of X is called data. A standard example is
when are i.i.d. random variables, In spatial
statistics, however, observations are typically dependent.
Let be a function of the parameter An estimator of
is a measurable function The quality of an estimator is
measured by its risk function
where Loss: is the loss function and denotes the expecta-
tion with respect to The bias of is given by
we say that is an unbiased estimator of if

In the following we consider the quadratic loss function
Then, if is unbiased we have
An unbiased estimator is a uniformly best unbiased estimator (UBUE)
of if
for any unbiased estimator of Thus a UBUE minimizes the risk
for all values (uniformly) among unbiased estimators.
Let be a Polish space. A measurable function is called
a sufficient statistic for the parameter if the conditional distribution
under the condition is independent of the parameter
In the dominated case (i.e., if the distributions are absolutely con-
tinuous with respect to some measure), the property of suffi-
ciency can be expressed by means of densities.
THEOREM 1.22 ([66]) Let the probability distributions have densi-
is sufficient for if and only if there exists a nonnega-
tive measurable function and a nonnegative measurable
function (independent of ) such that
for all
Further, a statistic is complete if the following implication
holds: If is a real measurable function on such that for
any then almost surely with respect to all distributions
THEOREM 1.23 (RAO-BLACKWELL) Let the experiment be
given, be a real parameter function on and let T be a sufficient
statistic for If is an arbitrary unbiased estimator of and T is
complete then
is a UBUE of The estimator is uniquelly determined a.s. in
the following sense: if is any unbiased estimator of with
for any then surely for any
and for all
ties on with respect to a measure A statistic

Preliminaries 19
REMARK 1.1. In view of the uniqueness assertion, we shall speak about
the UBUE estimator of a parameterfunction if the uniqueness a.s.
in the sense as described in the end of Theorem 1.23 holds.
The density of (cf. Theorem 1.22) taken for fixed as a
function of variable enables us to define the likelihood function
as Its maximum with respect to is called the
maximum likelihood estimator of Given the data
corresponding to a random sample X from the likelihood function is
factorized as
being the marginal densities of
In Bayesian statistics, the parameter is considered as a random
variable with prior distribution where is a fixed reference
Borel measure on Assume that data have been observed. The
posterior distribution is then the conditional distribution of given
and the Bayes estimator of is any number which
minimizes (with respect to the posterior risk
(this expectation is taken with respect to the prior distribution). For
the quadratic loss function (1.29), the Bayes estimator is the posterior
mean
Let be the likelihood function and a prior density. The
Bayes theorem yields the posterior density in the form
briefly
Statistical inference based on the posterior distribution requires eval-
uation of integrals Besides direct methods, the
MCMC approach (cf. Subsection 1.2.2) consists in an indirect evalua-
tion based on the simulation of the posterior. E.g., the posterior mean
of is estimated from
where is the generated Markov chain. For further applications, see
[34].
In the large sample theory, the sample X and estimator are
considered as functions of the sample size Such a sequence of

estimators of is said to be consistent if
for every The sequence is said to be strongly consistent if
almost sure convergence takes place in (1.33) instead of convergence in
probability. A sequence of Bayes estimators is called consistent if
condition (1.33) holds for all
One often speaks about the (strong) consistency of a single estimator
if its dependence on the size of data is clear. Finally consider
LEMMA 1.24 Let be a sequence of estimators of with risk
function
If
for all then is consistent, as well as the sequence of
Bayes estimators with the same risk function.
Proof. The first assertion follows from the fact that convergence in
probability is implied by the From (1.34) we have
when therefore surely and
the second assertion follows analogously to the first one.
LEMMA 1.25 Let be a compact parametric space and a
consistent sequence of estimators of the parameter Then (1.34) holds.
Proof. By the compactness,
is uniformly bounded by a constant M > 0. Since in probabil-
ity, following the proof of the Lebesgue dominated convergence theorem
applied to the random variable (1.35), the assertion follows.

Chapter 2
RANDOM MEASURES AND POINT PROCESSES
The purpose of this chapter is to survey basic facts about point pro-
cesses, processes of particles and associated random measures which will
be needed in the following chapters. The basic notion is that of a random
measure, a point process is a particular case of a random measure which
takes only integer values. This approach is rather unusual when dealing
with point processes on the real line, where we frequently interpret the
point process as a special case of a random function (a piecewise constant
function with jumps at the points of the process). In higher dimension
such an interpretation is no more possible. One can consider a point
process either as a locally finite collection of points (i.e., a special ran-
dom set), or as an integer valued random measure (measure with atoms
at the points of the process). Whereas the first approach seems to be
more illustrative and simpler, the second one has many technical advan-
tages when using the additivity of measures. We shall prefer the second
approach but we shall use the convention to interpret a point process
simultaneously as a collection of particles if this is more advantageous.
An outstandingly important tool in connection with point processes
and random measures is that of local conditioning known as the Palm
theory; we refer here namely to the monographs by Kallenberg [57],
Kerstan, Matthess and Mecke [60] and Daley and Vere-Jones [23]. Local
conditioning is, in fact, a special kind of disintegration.
Throughout the book, random structures generated as union sets of
processes of “particles” (which may be convex bodies, fibres, lines, flats
etc.) are considered. This chapter provides the necessary background for
these objects. The concepts of stationarity and isotropy are extremely
important here.

Besides of the references given above, we mention the monographs by
Stoyan, Kendall and Mecke [109] and Schneider and Weil [103] as basic
reference sources on random measures and point processes.
2.1. Basic definitions
Throughout this section, (X, is a Polish space which is locally com-
pact, i.e., to each there exists a neighbourhood with a compact
closure, and is the Borel on X. We denote by
the system of all closed, compact subsets of X,
respectively.
A measure on (X, is said to be locally finite if it is finite on
bounded Borel sets. By we denote the set of all locally
finite measures on (X, Further denote
the set of all locally finite integer-valued measures.
Let be the smallest on with respect to which the
function is measurable for all Further, let be the
trace of on i.e.,
We say that a sequence of measures converges vaguely to
if for each continuousfunction with compact
support.
The following result can be found e.g. in [23, Theorem A.2.6].
THEOREM 2.1 The space with the topology of vague convergence is
a Polish space and its Borel coincides with
DEFINITION 2.2 Let be a probability space. A random mea-
sure on X is a measurable mapping
A point process on X is a measurable mapping
The probability measure is the distribution of the ran-
dom measure (point process and the measure
is called the intensity measure of respectively). The point
process is simple if where
Note that the intensity measure need not be locally finite in general.

Random measures and point processes 23
Let us call a set locally finite if its intersection with an arbitrary
bounded set is finite.
REMARK 2.1. It is clear that almost all realizations of a simple point
process are characterized by their support supp which is a locally
finite subset of X. Therefore, simple point processes are often inter-
preted as locally finite random subsets of X. We shall sometimes use
this interpretation and write e.g. instead of or
THEOREM 2.3 ([23, PROPOSITIONS 7.1.II,III]) For each the
support supp is a locally finite subset of X. Further, and
is a one-to-one mapping of onto the set of all locally
finite subsets of X.
We shall demonstrate now the connection between simple point proces-
ses and random sets.
DEFINITION 2.4 A random closed set in X is a measurable mapping
where the on is generated by all families
It can be shown that the family of locally finite sets in X belongs to
(cf. Exercise 2.11).
THEOREM 2.5 If is a point process on X then supp is a random
closed set. On the other hand, if is a locally finite random closed set
in X (i.e., a random closed set in X such that is locally finite for
any then is a simple point process on X.
Proof. It is enough to verify the measurability of the mapping
from to locally finite} and of its inverse, which is
left to the reader as an exercise.
THEOREM 2.6 (CHOQUET, MATHERON) The distribution of a random
closed set is uniquelly determined by the probabilities
Proof. Note that the system is closed under finite
intersections. The result follows from the well known fact that a proba-
bility measure is uniquely determined by its values on a generator closed
w.r.t. finite intersections.

COROLLARY 2.7 The distribution of a simple point process is uniquely
determined by the “void probabilities”
DEFINITION 2.8 Let be a random measure on (X, with distribution
P and The measure
on is called the moment measure of order of Spe-
cially is the intensity measure of If is a point process and
we define also the factorial moment measure of order
where
of all of points of X with pairwise different coordinates.
REMARK 2.2. For two Borel sets A, we can write
and if is a simple point process then
and
using the convention explained in Remark 2.1.
EXERCISE 2.9 Verify the measurability of the mapping in the proof of
Theorem 2.5.
EXERCISE 2.10 Show that the families
and
is the restriction of to the set

belong to for any Borel set
EXERCISE 2.11 Show that the family of locally finite subsets of X be-
longs to Hint: Fix a bounded set and show that
To this end, use a sequence
of refining partitions of B into relatively compact sets with
diameters tending to 0 as and use the representation
EXERCISE 2.12 Show that for any nonnegative measurable function
on
EXERCISE 2.13 Show that
2.2. Palm distributions
THEOREM 2.14 (CAMPBELL) Let be a random measure on X with
distribution P and a locally finite intensity measure Then
for an arbitrary nonnegative measurable function on X. More gener-
ally, for and for any nonnegative measurable function on
we have
Proof. If is the characteristic function of a measurable set then the
results follow directly from the definitions. For nonnegative measurable
functions, we can use the standard approximation by simple functions
(see also [120, Theorem 5.2]).
DEFINITION 2.15 Let be a random measure on X with distribution P
and intensity measure The Campbell measure C corresponding to
is a measure on defined by

where is an arbitrary nonnegative measurable function on
Note that the Campbell measure C can also be characterized by the
property
where A is a bounded Borel subset of X and a measurable subset of
An important tool in the theory of random measures and point pro-
cesses are the Palm distributions which are‚ in fact‚ certain types of
conditional distributions. They are defined by means of a disintegration
of the Campbell measure as expressed in the following theorem (for its
proof see e.g. [23‚ Property 12.1.IV]).
THEOREM 2.16 Let be a random measure on X with distribution P
and a locally finite intensity measure Then there exists a probability
kernel from to such that
for an arbitrary nonnegative measurable function on If
is another probability kernel satisfying (2.5) then for any
measurable set
The distribution is called the Palm distribution of the random measure
at the point
REMARK 2.3. In fact, it has no sense to speak about the Palm distri-
bution in one particular point since this can be defined arbitrarily.
The uniqueness assertion from Theorem 2.16 nevertheless assures that
the family is uniquely determined for all
Let be the Palm distributions of the random measure
We shall sometimes use the notation for the Palm (conditional)
probability‚ which is formally defined as
Analogously‚ we shall write for the expectation with respect to the
Palm distribution at
In the case of a point process the Palm distribution can be
interpreted as the conditional distribution of under condition

(see [109‚ §4.4] or [57‚ Theorem 12.8]). In particular‚ the Campbell
measure C of a point process is concentrated on the set
Therefore‚ the following definition makes sense.
DEFINITION 2.17 If is a point process the reduced Campbell measure
is defined as
where is an arbitrary measurable function on The reduced
Palm distributions of are then defined again by means of
the disintegration
Sometimes‚ Palm distributions of higher order are needed. These can
be interpreted‚ in the case of a point process‚ as conditional distribu-
tions under condition that a given finite number of points belong to the
process. The formal definition is based again on the Campbell measure‚
now of a higher order.
Let be a random measure on X with distribution P and an intensity
measure and let be a natural number. The Campbell measure of
order corresponding to is a measure on defined by
where is an arbitrary nonnegative measurable function on
Assume now that the moment measure of is locally finite.
The Palm distributions of order are defined as a probability kernel
from to

If is a point process‚ the reduced Campbell measure oforder is
given by
and the reduced Palm distributions of order are determined
by
EXERCISE 2.18 Let be the point process in gener-
ated by independent identically distributed random vectors
Show that the Palm distribution of at is that of
2.3. Poisson process
The most familiar model of a point process is the Poisson process
which is introduced in analogy to the commonly known one-dimensional
case The basic property of the Poisson process is the mutual
independence of its behavior in disjoint domains.
DEFINITION 2.19 Let be a locally finite measure on a Polish space X‚
the system of bounded Borel subsets of X. A point process on X
such that
for any and disjoint‚ the random variables
are independent‚
has the Poisson distribution with parameter for any
1)
is called the Poisson process on X with intensity measure
The existence and uniqueness of the Poisson process follows from gen-
eral existence and uniqueness results on random measures‚ see [23‚ The-
orems 6.2.IV‚ 6.2.VII].
REMARK 2.4. It can be shown that if the intensity measure isdiffuse
(i.e.‚ if it has no atoms) then the corresponding Poisson process is simple‚
2)

see [23, §7.2]. According to Corollary 2.7, is in this case also uniquely
determined by the condition of being simple and the property
One can show directly from the definition that the factorial moment
measures of a Poisson process are product measures‚ i.e.‚
As a consequence one gets the followinglemma which will be used later
on. The proof follows from Theorem 2.14‚ Exercise 2.12 and (2.12).
LEMMA 2.20 ([82‚ LEMMA 2]) Let be a stationary Poisson point pro-
cess on a Polish space X with intensity measure Denote
for nonnegative measurable functions
Then
An important property of the Poisson process is that its reduced
Palm distribution coincides with its ordinary distribution. This fact
is known as the Slyvniak theorem‚ see e.g. [109‚ §4.4.6] or [23‚ Propo-
sition. 12.1.VI]. Recall that * denotes the convolution of measures‚ cf.
(1.2).
THEOREM 2.21 (SLYVNIAK) If P is the distribution of a Poisson point
process on X with locally finite intensity measure then
for all
Let be a random measure on X with distribution Q. A point process
on X is called a Cox process with driving measure if conditionally
on it is a Poisson process with intensity measure In other
words‚ the distribution of the Cox process is
where is the distribution of a Poisson process with intensity Of
course‚ the Cox process does not retain the independence property of
the Poisson process. The most common example is the Cox process in
with driving measure where Z is a positive random variable.

EXERCISE 2.22 Show that for the Poisson point process.
EXERCISE 2.23 Let be a Poisson process (not necessary stationary)
on a Polish space X with the intensity measure Let f be a measurable
function on X. Show that then
Hint: Start with the functions of the form where
and the are pairwise disjoint Borel subsets of X.
EXERCISE 2.24 Show that the void probabilities of the process are
EXERCISE 2.25 Let be the process with driving measure of
distribution Q and with locally finite intensity measure Then the
Palm distributions of are the mixtures
i.e.‚ is the distribution of a process with driving measure (the
Palm distribution of Q at
2.4. Finite point processes
Let be a Borel set‚ a measure on with
and P the distribution of a Poisson point process with intensity
measure Using Definition 2.19 and Exercise 2.29‚ we can write the
distribution P in the following way. Let be a set of (finite)
point configurations in X.
A point process has density on with respect to P if

A sufficient condition for the integrability of a nonnegative function
on with respect to P is its local stability‚ i.e.‚ existence of a constant
such that for all it holds
EXAMPLE 2.26 Let for a constant It holds
hence is P–integrable and using (2.15) we obtain
which is the normalizing constant for to become a probability density.
The distribution of the corresponding point process is
i. e.‚ is a Poisson point process with intensity measure
EXAMPLE 2.27 A frequent case is that X is bounded and is the
Lebesgue measure. Let be a measurable function on X. Then
is locally stable if and only if is bounded. In such a case we have
thus
is a probability density on Since

is a density of a Poisson point process with intensity function (density
of intensity measure w.r.t. Lebesgue measure)
EXAMPLE 2.28 The processes in Examples 2.26 and 2.27 are still of
Poisson type‚ they do not exhibit interactions among points. A simple
model with interactions‚ widely discussed in the literature‚ is the Strauss
process with density
where are parameters and
For it is a Poisson process‚ for there are repulsive in-
teractions. The limiting case is called a hard-core process‚ with
probability one there do not appear pairs ofpoints with distance less than
R. The Strauss process belongs to a large class of Markov point processes
[115] which are intensively studied.
EXERCISE 2.29 Show that the conditional distribution of a Poisson pro-
cess on X with finite intensity measure under condition
is that of a binomial distribution of i.i.d. points in X with distribution
Hint: The distribution of a point process (random measure) is
determined by its finite dimensional distributions of numbers of points
in pairvise disjoint sets (see [23, Proposition 6.2.III]). Let and let
be pairwise disjoint Borel subsets of X. Let be
nonnegative integers with and denote
Then, by using the Poisson property (Defini-
tion 2.19), show that
EXERCISE 2.30 Show that for the function in (2.17) is not
integrable and thus cannot serve as a probability density.
2.5. Stationary random measures on
Let be a random measure on We shall denote for brevity
For let denote the corresponding shift
operator on defined by

The random measure is called stationary if its distribution is shift
invariant‚ i.e.‚ if has the same distribution as for any
Further‚ given a rotation we define the corresponding
rotation operator on by
The random measure is called isotropic if its distribution is invariant
under for any
A well-known measure-theoretic fact implies that ifthe intensity mea-
sure of a stationary random measure is locally finite then it is a
multiple of the Lebesgue measure‚ say The constant is
called the intensity of the stationary random measure
The stationarity implies obviously the shift covariance of the Palm
distributions. The following theorem presents a possible choice of the
Palm distributions for a stationary random measure.
THEOREM 2.31 A stationary random measure on with intensity
has Palm distributions
where A is an arbitrary Borel set in with positive and finite Lebesgue
measure.
To prove the theorem‚ one has to verify that the family ofdistributions
satisfies (2.5)‚ see also [23‚ Theorem 12.2.II].
Theorem 2.31 provides an explicit formula for calculating the Palm
distribution at the origin. Whenever talking about the Palm distribution
at the origin of a stationary random measure‚ we shall always mean
by this that is a family of Palm distributions of the
random measure.
Some further characteristics are often used for stationary random mea-
sures. The reduced second moment measure is defined by
Note that if‚ in particular‚ is a point process we have

If there exists a density of . w.r.t. is called the pair correlation
function. The K-function is defined by
If is a stationary point process and the reduced Palm distribution
it holds
Thus in the stationary case the second order moment mea-
sures can be expressed by means of and
Consider again a stationary random measure with intensity If
we can write by using Theorem 2.31
with an arbitrary bounded Borel set A of positive Lebesgue measure.
Let now be two stationary random measures with intensities
respectively. Assume that are jointly stationary‚ i.e.‚ that
the joint distribution of is the same as that of for
any shift In analogy to (2.21)‚ we define the cross correlation
measure of and as
(the set A is as above)‚ cf. [110‚ 111]. The cross-correlation function
of and is then the density of w.r.t. Lebesgue measure
(if it exists).
REMARK 2.5. If two random measures are independent‚ their
cross-correlation measure is the Lebesgue measure and‚ hence‚
for (almost) all
EXERCISE 2.32 Show that the pair correlation function of a stationary
Poisson process is If is a stationary Cox process with
driving measure (i.e., a stationary “Poisson” process with random

intensity Z)‚ its intensity is EZ and pair correlation function is again
constant‚
EXERCISE 2.33 Let be a stationary Gaussian
random field (almost surely continuous) with mean variance and
correlation function The Cox process with driving measure
is called a log-Gaussian Cox process (LGCP). Show that
the factorial measure of a LGCP on has a density (called
a product density) w.r.t. of form
The distribution of a LGCP is determined by the intensity
and the pair correlation function (Hint: Use
Corollary 2.7.)
EXERCISE 2.34 Show that any two jointly stationary random measures
in with intensities fulfill
for any Further‚ if the cross-correlation functions and
exist they are symmetric in the sense that
cf. [110].
2.6. Application of point processes in
epidemiology
The aim of statistical disease mapping is to characterize the spatial
variation of cases of a disease and to study connections with covari-
ates. In the present example tick-borne encephalitis (TBE)‚ an infection
illness which is transmitted by parasitic ticks and which occasionally
afflicts humans‚ is a disease in question. Epidemiologists and medical
practitioners making decision on prophylactic measures deal with the
problem of estimating the risk that a human gets infected by TBE at
a specific location. Usually the data for statistical analysis consist of
case locations and a population map. Moreover‚ explanatory variables
ofgeographical nature which may influence the risk ofinfection are often
given from geographical information systems.
The data were collected by Zeman [127]. A point pattern of locations
of 446 reported cases of TBE in Central Bohemia (region denoted by

S in the following) during 1971-93 is available‚ see Fig. 2.1. Different
covariates are considered: the locations of forests of areas between 10-50
and 50-150 ha‚ respectively‚ the subareas of three different forest types
(conifer‚ foliate‚ and mixed forest) and a map of altitudes.
Finally‚ population data for the Central Bohemia consist of the num-
ber of inhabitants in 3582 administrative units.
The modelling of the TBE data in [7] is motivated by the following
simplifying considerations. In the observation period 1971-93 a number
of inhabitants are living at home locations
S‚ and the person makes a number of visits to the surroundings
of The are assumed to be independent and Poisson distributed
with mean independent of Given the the location of each
visit of the person is distributed according to some density (with
respect to and the locations of visits of all persons are assumed
to be independent. For a visit to a location there is associated
a probability for getting an infection during the visit. The point
process of locations where persons have been infected (cases) is then a
Poisson process with intensity function
where is the background intensityofhumans visiting
We model in (2.23) by a log linear model,
where is a zero-mean Gaussian process‚ is a
regression parameter‚ and Here is an
intercept‚ and are six covariates associated with the lo-
cation where the index corresponds to the following:
1 ~ forest 10-50 ha‚ 2 ~ forest 50-150 ha‚ 3 ~ conifer forest‚
4 ~ mixed forest‚ 5 ~ foliate forest‚ 6 ~ altitude.
Here are 0-1 functions (equal to 1 in the case of presence of
the characteristics). The role of is partly to model deviations
of from one‚ being an estimator of unknown Therefore we
do not constrain (2.24) to be less than one‚ actually‚ is absorbed in
Then is more precisely a relative risk function.
The process Y is assumed to be second-order stationary and isotropic
with exponential covariance function‚ i.e.‚

where is the variance and is the correlation parameter.
A log-Gaussian Cox process is then obtained by assuming that condi-
tionally on and the cases form a Poisson
process with intensity function
A hierarchical Bayesian approach is adapted (cf. [34]). The Gaussian
distribution for Y is viewed as a prior and the conditional distribution of
given as the likelihood. Furthermore‚ a hyper prior density
for is imposed; specific hyper priors are considered. The likelihood
is derived from the density with respect to the unit rate Poisson process
on S‚ cf. (2.16).
The posterior‚ that is‚ the conditional distribution of given
can be specified as follows. Suppose that is proper and let
denote expectation conditionally on For and pairwise dis-
tinct let denote the conditional density of
given The posterior density of
given is defined by
The posterior is then given by the consistent set of finite-dimensional
posterior distributions with densities of the form (2.27) for and
pairwise distinct The integral in (2.26) depends on the
continuous random field Y which cannot be represented on a computer.
In practice the integral is approximated by a Riemann sum. The aim
is an MCMC simulation of the approximate posterior when
agrees with the set of centres of squares of a lattice covering S with
size M × M.
The main obstacle is to handle the high dimensional covariance matrix
of However‚ the computational cost can be reduced
very much by employing the circulant embedding technique described in
[26] and [124]; see also [79]. For the MCMC simulations of given
a hybrid algorithm as described in [80] was used‚ where
and are updated in turn using so-called truncated Langevin-Hastings
updates for and standard random walk Metropolis updates for and
Geometric ergodicity is thus achieved.
The posterior mean of the relative risk function is plotted as a result of
analysis with M = 64‚ see Fig. 2.1. In [7] several choices of background

Other documents randomly have
different content

the trees, but they were in such inaccessible ravines that the
shipbuilder declared it was impossible to get them.
Ugarte was not swayed from his purpose by this difficulty; he
went down to Loretto and returned with three mechanics and all the
Indians he could induce to follow him. After four months of hard
work he not only had all the trees felled and shaped, but he had
opened a road for thirty leagues over the mountains and with oxen
and mules hauled his material to the coast. He built his "Triumph of
the Cross," as he called it, in four months. The provincial was told
meanwhile, that it was going to be used for pearl fishing, and sent
the supposed culprit a very sharp letter in consequence. No doubt
he made amends for this when he was disabused. The "Triumph of
the Cross" was not to carry a cargo of pearls but was intended to
explore the upper Gulf, so as to realize the dream of Kino and
Salvatierra.
The good ship left Loretto on May 15, 1721, with twenty men,
six of whom were Europeans, the captain being a William Stafford. It
was followed by the "Santa Barbara," a large open boat carrying five
Californians, two Chinese and a Yaqui. They made their first landing
at Concepción Bay, and then, after creeping along the shore
northward, crossed the Gulf to Santa Sabina and San Juan Bautista
on the Seri coast. The sight of the cross on the bow-sprit delighted
the natives and assured the travellers of a hearty welcome. Tiburon
was the next stop, and while there Ugarte felt his strength giving
out; but despite his sixty-one years he continued his voyage, and
headed the "Triumph" for the mouth of the Colorado, while the
"Santa Barbara" hugged the shore. Meantime, a few men were
landed and made for the nearest mission. They found the trail to
Caborca and soon the Jesuits of that place and of San Ignacio
hurried down with provisions for the travellers.
While the "Santa Barbara" was being loaded, the "Triumph" was
nearly stranded at the mouth of the river, so it was decided to cross
to the other side, which they reached only after a hard three days'
sail. There the "Santa Barbara" met them and both ships pointed

north, crossing and recrossing the gulf until finally they anchored at
the mouth of the river on the Pimería side. There was some talk of
going up the stream, but the ship's position in the strong current
was dangerous, the weather was threatening, and besides, Ugarte
had achieved his purpose; he had seen the river from the Gulf and
had added a convincing proof to Kino's assertion that California was
a peninsula. On July 16 they started south; the storm they had
feared broke over them and the sloop nearly went to the bottom.
The sailors, who were nearly all sick of the scurvy, got confused in
the Salsipuedes channel, and it was only on August 18 that they
cleared that passage so aptly called "Get out if you can." But a triple
rainbow in the sky that day comforted them, just as they had been
cheered when the St. Elmo's fire played around the mast head
during the gale. But they were not free yet. Another storm overtook
them and they had great difficulty in dodging a waterspout, but they
finally reached Loretto in the month of September.
Besides its original purpose, this voyage resulted in furnishing
much valuable information about the shores, ports, islands and
currents of the Upper Gulf. The original account of the journey with
maps and a journal kept by Stafford was sent to the viceroy for the
king, but Bancroft says they have not been traced. Ugarte lived only
eight years after this eventful journey. Picolo, Salvatierra's first
companion had preceded him to the grave, dying on February 22,
1729, at the age of 79, whereas Ugarte's life-work did not cease till
the following December 29. Perhaps Lower California owes more to
him than to the great Salvatierra.
A classic example of the influence of ignorance in the creation of
many of the false statements of history is furnished by a publication
about these missions in the "Montreal Gazette" of 1847, under the
title of "Memories of Mgr. Blanchet." "The failure of the Jesuits in
Lower California," he says, "must be attributed to their unwillingness
to establish a hierarchy in that country. Had they been so disposed,
they might have had a metropolitan and several suffragans on the
Peninsula. They failed to do so, until at last, in 1767, word came
from generous Spain to hand over their work to some one else." In

the first place, "generous Spain" had not the slightest desire to
establish a hierarchy on that barren neck of land when it expelled
the Jesuits in 1767. Again as "generous Spain" appointed even the
sacristans in its remotest colonies, the Society must be acquitted of
all blame in not giving an entire hierarchy to Lower California.
Finally, one hundred and fifty-one years have elapsed since the last
Jesuits left both Mexico and Lower California and there is nothing
there yet, but the little Vicariate Apostolic of La Paz down at the
lower end of the Peninsula.
In describing the work of the Jesuits in Mexico, Bancroft (XI,
436) writes as follows: "Without discussing the merits of the charges
preferred against them, it must be confessed that the service of God
in their churches was reverent and dignified. They spread education
among all classes, their libraries were open to all, and they
incessantly taught the natives religion in its true spirit, as well as the
mode of earning an honest living. Among the most notable in the
support of this last assertion are those of Nayarit, Sonora, Sinaloa,
Chihuahua and lower California, where their efforts in the conversion
of the natives were marked by perseverance and disinterestedness,
united with love for humanity and prayer. Had the Jesuits been left
alone, it is doubtful whether the Spanish-American province would
have revolted so soon, for they were devoted servants of the crown
and had great influence with all classes — too great to suit royalty,
but such as after all might have saved royalty in these parts."
Indeed, when the Society was re-established in 1814, Spain had
already lost nearly all of its American colonies. The punishment had
rapidly followed the crime.
Although Mexico and the Philippines are geographically far apart,
yet ecclesiastically one depended on the other. Legaspi, who took
possession of the islands in 1571, built his fleet in Mexico, and also
drafted his sailors there. Andrés de Urdaneta, the first apostle of the
Philippines, was an Augustinian friar in Mexico who accompanied
Legaspi as his chaplain. Twenty years after that expedition, the
Jesuits built their first house in Manila, and Father Sánchez, who
was, as we have said, one of the supervisors of the great tunnel,

was sent as superior from Mexico to Manila. One of his companions,
Sedeño, had been a missionary in Florida, and it was he who opened
the first school in the Philippines and founded colleges at Manila and
Cebú. He taught the Filipinos to cut stone and mix mortar, to weave
cloth and make garments. He brought artists from China to teach
them to draw and paint, and he erected the first stone building in
the Philippines, namely the cathedral, dedicated in honor of the
Immaculate Conception of the Blessed Virgin. His religious superior,
Father Sánchez had meanwhile acquired such influence in Manila as
to be chosen in 1585, by a unanimous vote of all the colonists, to go
to arrange the affairs of the colony with Philip II and the Pope. He
brought with him to Europe a Filipino boy who, on his return to his
native land, entered the Society, and became thus the first Filipino
Jesuit.
The college and seminary of San José was established in Manila
in 1595. It still exists, though it is no longer in the hands of the
Society; being the oldest of the colleges of the Archipelago, it was
given by royal decree precedence over all other educational
institutions. During the first hundred years of its educational life, it
counted among its alumni, eight bishops and thirty-nine Jesuits, of
whom four became provincials. There were also on the benches
eleven future Augustinians, eighteen Franciscans, three Dominicans,
and thirty-nine of the secular clergy. The University of St. Ignatius,
which opened its first classes in 1587, was confirmed as a pontifical
university in 1621 and as a royal university in 1653. Besides these
institutions, the Society had a residence at Mecato and a college at
Cavite, and also the famous sanctuary of Antipole. They likewise
established the parishes of Santa Cruz and San Miguel in Manila.
France began its colonization in North America by the settlement
of Acadia in 1603. De Monts, who was in charge of it, was a
Huguenot and, strange to say, had been commissioned to advance
the interests of Catholicity in the colony. Half of the settlers were
Calvinists, and the other half Catholics more or less infected with
heresy. A priest named Josué Flesché was assigned to them; he
baptized the Indians indiscriminately, letting them remain as fervent

polygamists as they were before. The two Jesuit missionaries, Pierre
Biard and Enemond Massé, who were finally forced on the colonists,
had to withdraw, and they then betook themselves, in 1613, to what
is now known as Mount Desert, in the state of Maine, but that
settlement was almost immediately destroyed by an English pirate
from Virginia. Two of the Jesuits were sentenced to be hanged in the
English colony there, but thanks to a storm which drove them across
the Atlantic, they were able, after a series of romantic adventures, to
reach France, where they were accused of having prompted the
English to destroy the French settlement of Acadia.
Meantime, Champlain, who had established himself at Quebec in
1608, brought over some Recollect Friars in 1615. It was not until
1625 that Father Massé, who had been in Acadia, came to Canada
proper with Fathers de Brébeuf, Charles Lalemant, and two lay-
brothers. With the exception of Brébeuf, they all remained in
Quebec, while he with the Recollect La Roche d'Aillon went to the
Huron country, in the region bordering on what is now Georgian Bay,
north of the present city of Toronto. The Recollect returned home
after a short stay, and Brébeuf remained there alone until the fall of
Quebec in 1629. As the English were now in possession, all hope of
pursuing their missionary work was abandoned, and the priests and
brother returned to France. Canada, however, was restored to its
original owners in 1632, and Le Jeune and Daniel, soon to be
followed by Brébeuf and many others, made their way to the Huron
country to evangelize the savages. The Hurons were chosen because
they lived in villages and could be more easily evangelized, whereas
the nomad Algonquins would be almost hopeless for the time being.
The Huron missions lasted for sixteen years. In 1649 the tribe
was completely annihilated by their implacable foes, the Iroquois, a
disaster which would have inevitably occurred, even if no missionary
had ever visited them. The coming of the Jesuits at that particular
time seemed to be for nothing else than to assist at the death
agonies of the tribe. The terrible sufferings of those early
missionaries have often been told by Protestant as well as Catholic
writers. At one time, when expecting a general massacre, they sat in

their cabin at night and wrote a farewell letter to their brethren; but,
for some reason or other, the savages changed their minds, and the
work of evangelization continued for a little space. Meantime,
Brébeuf and Chaumonot had gone down as far as Lake Erie in mid-
winter and, travelling all the distance from Niagara Falls to the
Detroit River, had mapped out sites for future missions. Jogues and
Raymbault, setting out in the other direction, had gone to Lake
Superior to meet some thousands of Ojibways who had assembled
there to hear about "the prayer."
The first great disaster occurred on August 3, 1642. Jogues was
captured near Three Rivers, when on his way up from Quebec with
supplies for the starving missionaries. He was horribly mutilated, and
carried down to the Iroquois country, where he remained a prisoner
for thirteen months, undergoing at every moment the most terrible
spiritual and bodily suffering. His companion, Goupil was murdered,
but Jogues finally made his escape by the help of the Dutch at
Albany, and on reaching New York was sent across the ocean in mid-
winter, and finally made his way to France. He returned, however, to
Canada, and in 1644 was sent back as a commissioner of peace to
his old place of captivity. It was on this journey that he gave the
name of Lake of the Blessed Sacrament to what is called Lake
George. In 1646 he returned again to the same place as a
missionary, but he and his companion Lalande were slain; the reason
of the murder being that Jogues was a manitou who brought
disaster on the Mohawks. Two other Jesuits, Bressani and Poncet,
were cruelly tortured at the very place where Jogues had been slain,
but were released.
In 1649 the Iroquois came in great numbers to Georgian Bay to
make an end of the Hurons. Daniel, Gamier and Chabanel were
slain, and Brébeuf and Lalemant were led to the stake and slowly
burned to death. During the torture, the Indians cut slices of flesh
from the bodies of their victims, poured scalding water on their
heads in mockery of baptism, cut the sign of the cross on their flesh,
thrust red-hot rods into their throats, placed live coals in their eyes,
tore out their hearts, and ate them, and then danced in glee around

the charred remains. This double tragedy of Brébeuf and Lalemant
occurred on the 16th and 17th of March, 1649. After that the Hurons
were scattered everywhere through the country, and disappeared
from history as a distinct tribe.
As early as 1650 there was question of a bishop for Quebec. The
queen regent, Anne of Austria, the council of ecclesiastical affairs,
and the Company of New France all wrote to the Vicar-General of
the Society asking for the appointment of a Jesuit. The three Fathers
most in evidence were Ragueneau, Charles Lalemant and Le Jeune.
All three had refused the honor and Father Nickel wrote to the
petitioners that it was contrary to the rules of the Order to accept
such ecclesiastical dignities. The hackneyed accusation of the
supposed Jesuit opposition to the establishment of an episcopacy
was to the fore even then in America. The refutation is handled in a
masterly fashion by Rochemonteix (Les Jésuites et la Nouvelle
France, I, 191). Incidentally the prevailing suspicion that Jesuits are
continually extolling each other will be dispelled by reading the
author's text and notes upon the characteristics of the three
nominees which unfitted them for the post. "Le Jeune," he says,
"would be unfit because he was a converted Protestant who had
never rid himself of the defects of his early education." It was not
until 1658 that Laval was named.
Meantime in 1654, through the efforts of Father Le Moyne to
whom a monument has been erected in the city of Syracuse, a line
of missions was established in the very country of the Iroquois. It
extended all along the Mohawk from the Hudson to Lake Erie. Many
of the Iroquois were converted such as Garagontia, Hot Ashes and
others, the most notable of whom was the Indian girl, Tegakwitha,
who fled from the Mohawk to Caughnawaga, a settlement on the St.
Lawrence opposite Lachine which the Fathers had established for the
Iroquois converts. The record of her life gives evidence that she was
the recipient of wonderful supernatural graces. These New York
missions were finally ruined by the stupidity and treachery of two
governors of Quebec, de la Barre and de Denonville, and also by the
Protestant English who disputed the ownership of that territory with

the French. By the year 1710 there were no longer any missionaries
in New York, except an occasional one who stole in, disguised as an
Indian, to visit his scattered flock. There were three Jesuits with
Dongan, the English governor of New York during his short tenure of
office, but they never left Manhattan Island in search of the Indians.
Attention was then turned to the Algonquins, and there are
wonderful records of heroic missionary endeavor all along the St.
Lawrence from the Gulf to Montreal, and up into the regions of the
North. Albanel reached Hudson Bay, and Buteux was murdered at
the head-waters of the St. Maurice above Three Rivers. The Ottawas
in the West were also looked after, and Garreau was shot to death
back of Montreal on his way to their country, which lay along the
Ottawa and around Mackinac Island and in the region of Green Bay.
The heroic old Ménard perished in the distant swamps of Wisconsin;
Allouez and Dablon travelled everywhere along the shores of Lake
Superior; a great mission station was established at Sault Ste. Marie,
and Marquette with his companion Joliet went down the Mississippi
to the Arkansas, and assured the world that the Great River emptied
its waters in the Gulf of Mexico. A statue in the Capitol of
Washington commemorates this achievement and has been
duplicated elsewhere.
The beatification of Jogues, Brébeuf, Lalemant, Daniel, Gamier,
Chabanel and the two donnés, Goupil and Lalande, is now under
consideration at Rome. Their heroic lives as well as those of their
associates have given rise to an extensive literature, even among
Protestant writers, but the most elaborate tribute to them is
furnished by the monumental work consisting of the letters sent by
these apostles of the Faith to their superior at Quebec and known
the world over as "The Jesuit Relations." It comprises seventy-three
octavo volumes, the publication of which was undertaken by a
Protestant company in Cleveland. (See Campbell, Pioneer Priests of
North America.)
On March 25, 1634, the Jesuit Fathers White and Altham landed
with Leonard Calvert, the brother of Lord Baltimore, on St. Clement's

Island in Maryland. With them were twenty "gentlemen
adventurers," all of whom, with possibly one exception, were
Catholics. They brought with them two hundred and fifty mechanics,
artisans and laborers who were in great part Protestants. It took
them four months to come from Southampton and, on the way over,
all religious discussions were prohibited. They were kindly received
by the Indians, and the wigwam of the chief was assigned to the
priests. A catechism in Patuxent was immediately begun by Father
White, and many of the tribe were converted to the Faith in course
of time, as were a number of the Protestant colonists. Beyond that,
very little missionary work was accomplished, as all efforts in that
direction were nullified by a certain Lewger, a former Protestant
minister who was Calvert's chief adviser. The adjoining colony of
Virginia, which was intensely bitter in its Protestantism, immediately
began to cause trouble. In 1644 Ingle and Claiborne made a descent
on the colony in a vessel, appropriately called the "Reformation."
They captured and burned St. Mary's, plundered and destroyed the
houses and chapels of the missionaries, and sent Father White in
chains to England, where he was to be put to death, on the charge
of being "a returned priest." As he was able to show that he had
"returned" in spite of himself, he was discharged.
Calvert recovered his possessions later, and then dissensions
began between him and the missionaries because of some land
given to them by the Indians. In 1645 it was estimated that the
colonists numbered between four and five thousand, three-fourths of
whom were Catholics. They were cared for by four Jesuits. In 1649
the famous General Toleration Act was passed, ordaining that "no
one believing in Jesus Christ should be molested in his or her
religion." As the reverse of this obtained in Virginia, at that time, a
number of Puritan recalcitrants from that colony availed themselves
of the hospitality of Maryland, and almost immediately, namely in
1650, they repealed the Act and ordered that "no one who professed
and exercised the Papistic, commonly known as the Roman Catholic
religion, could be protected in the Province." Three of the Jesuits
were, in consequence, compelled to flee to Virginia, where they kept

in hiding for two or three years. In 1658 Lord Baltimore was again in
control, and the Toleration Act was re-enacted. In 1671 the
population had increased to 20,000, but in 1676 there was another
Protestant uprising and the English penal laws were enforced against
the Catholic population. In 1715 Charles, Lord Baltimore, died.
Previous to that, his son Benedict had apostatized and was
disinherited. He died a few months after his father. Benedict's son
Charles, who was also a turncoat, was named lord proprietor by
Queen Ann, and made the situation so intolerable for Catholics that
they were seriously considering the advisability of abandoning
Maryland and migrating in a body to the French colony of Louisiana.
As a matter of fact many went West and established themselves in
Kentucky.

Of the Jesuits and their flock in Maryland, Bancroft writes: "A
convention of the associates for the defence of the Protestant
religion assumed the government, and in an address to King William
denounced the influence of the Jesuits, the prevalence of papist
idolatry, the connivances of the previous government at murders of
Protestants and the danger from plots with the French and Indians.
The Roman Catholics in the land which they had chosen with
Catholic liberality, not as their own asylum only, but as the asylum of
every persecuted sect, long before Locke had pleaded for toleration,
or Penn for religious freedom, were the sole victims of Protestant
intolerance. Mass might not be said publicly. No Catholic priest or
bishop might utter his faith in a voice of persuasion. No Catholic
might teach the young. If the wayward child of a Catholic would
become an apostate the law wrested for him from his parents a
share of their property. The disfranchisement of the Proprietary
related to his creed, not to his family. Such were the methods
adopted to prevent the growth of Popery. Who shall say that the
faith of the cultivated individual is firmer than the faith of the
common people? Who shall say that the many are fickle; that the
chief is firm? To recover the inheritance of authority Benedict, the
son of the Proprietary, renounced the Catholic Church for that of
England, but the persecution never crushed the faith of the humble
colonists."
The extent of the Jesuit missions in what is now Canada and the
United States may be appreciated by a glance at the remarkable
map recently published by Frank F. Seaman of Cleveland, Ohio. On it
is indicated every mission site beginning with the Spanish posts in
Florida, Georgia and Virginia, as far back as 1566. The missions of
the French Fathers are more numerous, and extend from the Gulf of
Mexico to Hudson Bay, and west to the Great Lakes and the
Mississippi. Not only are the mission sites indicated, but the habitats
of the various tribes, the portages and the farthest advances of the
tomahawk are there also. Lines starting from Quebec show the
source of all this stupendous labor.

CHAPTER XI
CULTURE
Colleges — Their Popularity — Revenues — Character of education: Classics;
Science; Philosophy; Art — Distinguished Pupils — Poets: Southwell; Balde;
Sarbievius; Strada; Von Spee; Gresset; Beschi. — Orators: Vieira; Segneri;
Bourdaloue. — Writers: Isla; Ribadeneira; Skarga; Bouhours etc. — Historians —
Publications — Scientists and Explorers — Philosophers — Theologians — Saints.
To obviate the suspicion of any desire of self-glorification in the
account of what the Society has achieved in several fields of
endeavor especially in that of science, literature and education it will
be safer to quote from outside and especially from unfriendly
sources. Fortunately plenty of material is at hand for that purpose.
Böhmer-Monod, for instance, in "Les Jésuites" are surprisingly
generous in enumerating the educational establishments possessed
by the Society at one time all over Europe, though their explanation
of the phenomenon leaves much to be desired. In 1540, they tell us,
"the Order counted only ten regular members, and had no fixed
residence. In 1556 it had already twelve provinces, 79 houses, and
about 1,000 members. In 1574 the figures went up to seventeen
provinces, 125 colleges, 11 novitiates, 35 other establishments of
various kinds, and 4,000 members. In 1608 there were thirty-one
provinces, 306 colleges, 40 novitiates, 21 professed houses, 65
residences and missions, and 10,640 members. Eight years
afterwards, that is a year after the death of its illustrious General
Aquaviva, the Society had thirty-two provinces, 372 colleges, 41
novitiates, 123 residences, 13,112 members. Ten years later, namely
in 1626, there were thirty-six provinces, 2 vice-provinces, 446
colleges, 37 seminaries, 40 novitiates, 24 professed houses, about
230 missions, and 16,060 members. Finally in 1640 the statistics
showed thirty-five provinces, 3 vice-provinces, 521 colleges, 49

seminaries, 54 novitiates, 24 professed houses, about 280
residences and missions and more than 16,000 members."
Before giving these "cold statistics," as they are described, the
authors had conducted their readers through the various countries of
Europe, where this educational influence was at work. "Italy," we are
informed, "was the place in which the Society received its
programme and its constitution, and from which it extended its
influence abroad. Its success in that country was striking, and if the
educated Italians returned to the practices and the Faith of the
Church, if it was inspired with zeal for asceticism and the missions, if
it set itself to compose devotional poetry and hymns of the Church,
and to consecrate to the religious ideal, as if to repair the past, the
brushes of its painters and the chisels of its sculptors, is it not the
fruit of the education which the cultivated classes received from the
Jesuits in the schools and the confessionals? Portugal was the
second fatherland of the Society. There it was rapidly acclimated.
Indeed, the country fell, at one stroke, into the hands of the Order;
whereas Spain had to be won step by step. It met with the
opposition of Spanish royalty, the higher clergy, the Dominicans.
Charles V distrusted them; Philip II tried to make them a political
machine, and some of the principal bishops were dangerous foes,
but in the seventeenth century the Society had won over the upper
classes and the court, and soon Spain had ninety-eight colleges and
seminaries richly endowed, three professed houses, five novitiates,
and four residences, although the population of the country at that
time was scarcely 5,000,000.
"In France a few Jesuit scholars presented themselves at the
university in the year 1540. They were frowned upon by the courts,
the clergy, the parliament, and nearly all the learned societies. It
was only in 1561, after the famous Colloque de Poissy, that the
Society obtained legal recognition and was allowed to teach, and in
1564 it had already ten establishments, among them several
colleges. One of the colleges, that of Clermont, became the rival of
the University of Paris, and Maldonatus, who taught there, had a
thousand pupils following his lectures. In 1610 there were five

French provinces with a total of thirty-six colleges, five novitiates,
one professed house, one mission, and 1400 members. La Flèche,
founded by Henry IV, had 1,200 pupils. In 1640 the Society in France
had sixty-five colleges, two academies, two seminaries, nine
boarding-schools, seven novitiates, four professed houses, sixteen
residences and 2050 members.
"In Germany Canisius founded a boarding school in Vienna, with
free board for poor scholars, as early as 1554. In 1555 he opened a
great college in Prague; in 1556, two others at Ingolstadt and
Cologne respectively, and another at Munich in 1559. They were all
founded by laymen, for, with the exception of Cardinal Truchsess of
Augsburg, the whole episcopacy was at first antagonistic to the
Order. In 1560 they found the Jesuits their best stand-by, and in
1567 the Fathers had thirteen richly endowed schools, seven of
which were in university cities. The German College founded by
Ignatius in Rome was meantime filling Germany with devoted and
learned priests and bishops, and between 1580 and 1590
Protestantism disappeared from Treves, Mayence, Augsburg,
Cologne, Paderborn, Münster and Hildesheim. Switzerland gave
them Fribourg in 1580, while Louvain had its college twenty years
earlier.
"In 1556 eight Fathers and twelve scholastics made their
appearance at Ingolstadt in Bavaria. The poison of heresy was
immediately ejected, and the old Church took on a new life. The
transformation was so prodigious that it would seem rash to
attribute it to these few strangers; but their strength was in inverse
proportion to their number. They captured the heart and the head of
the country, from the court and the local university down to the
people; and for centuries they held that position. After Ingolstadt
came Dillingen and Würzburg. Munich was founded in 1559, and in
1602 it had 900 pupils. The Jesuits succeeded in converting the
court into a convent, and Munich into a German Rome. In 1597 they
were entrusted with the superintendence of all the primary schools
of the country, and they established new colleges at Altoetting and
Mindelheim. In 1621 fifty of them went into the Upper Palatinate,

which was entirely Protestant, and in ten years they had established
four new colleges.
"In Styria, Carinthia, and Carniola there was scarcely a vestige of
the old Church in 1571. In 1573 the Jesuits established a college at
Grätz, and the number of communicants in that city rose
immediately from 20 to 500. The college was transformed into a
university twelve years later, and in 1602 and 1613 new colleges
were opened at Klagenfurth and Leoben. In Bohemia and Moravia
they had not all the secondary schools, but the twenty colleges and
eleven seminaries which they controlled in 1679 proved that at least
the higher education and the formation of ecclesiastics was
altogether in their hands, and the seven establishments and colleges
on the northern frontier overlooking Lutheran Saxony made it
evident that they were determined to guard Bohemia against the
poison of heresy." The writer complains that they even dared to
dislodge "Saint John Huss" from his niche and put in his place St.
John Nepomucene, "who was at most a poor victim, and by no
means a saint." Böhmer's translator, Monod, adds a note here to
inform his readers that the Jesuits invented the legend about St.
John Nepomucene, and induced Benedict XIII to canonize him.
Finally, we reach Poland where, we are informed that "the
Jesuits enjoyed an incredible popularity. In 1600 the college of
Polotsk had 400 students, all of whom were nobles; Vilna had 800,
mostly belonging to the Lithuanian nobility, and Kalisch had 500.
Fifty years later, all the higher education was in the hands of the
Order, and Ignatius became, literally, the preceptor Poloniæ, and
Poland the classic land of the royal scholarship of the north, as
Portugal was in the south.
"In India, there were nineteen colleges and two seminaries; in
Mexico, fourteen colleges and two seminaries; in Brazil, thirteen
colleges and two seminaries; in Paraguay, seven colleges," and the
authors might have added, there was a college in Quebec, which
antedated the famous Puritan establishment of Harvard in New
England, and which was erected not "out of the profits of the fur

trade," as Renaudot says in the Margry Collection, but out of the
inheritance of a Jesuit scholastic.
After furnishing their readers with this splendid list of houses of
education, the question is asked: "How can we explain this incredible
success of the Order as a teaching body? If we are to believe the
sworn enemies of the Jesuits, it is because they taught gratuitously,
and thus starved out the legitimate successors of the Humanists.
That might explain it somewhat, they say, especially in southern
Italy, where the nobleman is always next door to the lazzarone, but
it will by no means explain how so many princes and municipalities
made such enormous outlays to support those schools; for there
were other orders in Catholic countries as rigidly orthodox as the
Jesuits. No; the great reason of their success must be attributed to
the superiority of their methods. Read the pedagogical directions of
Ignatius, the great scholastic ordinances of Aquaviva, and the
testimony of contemporaries, and you will recognize the glory of
Loyola as an educator. The expansion is truly amazing; from a
modest association of students to a world-wide power which ended
by becoming as universal as the Church for which it fought; but
superior to it in cohesion and rapidity of action — a world power
whose influence made itself felt not only throughout Europe, but in
the New World, in India, China, Japan; a world power on whose
service one sees at work, actuated by the same spirit,
representatives of all races and all nations: Italians, Spaniards,
Portuguese, French, Germans, English, Poles and Greeks, Arabians,
Chinamen and Japanese and even red Indians; a world power which
is something such as the world has never seen."
Another explanation is found in the vast wealth which "from the
beginning was the most important means employed by the Order."
We are assured that the Jesuits have observed on this point such an
absolute reserve that it is still impossible to write a history or draw
up an inventory of their possessions. But, perhaps it might be
answered that if an attempt were also made to penetrate "the
absolute reserve" of those who have robbed the Jesuits of all their
splendid colleges and libraries and churches and residences which

may be seen in every city of Europe and Spanish America, with the
I.H.S. of the Society still on their portals, some progress might be
made in at least drawing up an inventory of their possessions.
As a matter of fact the Jesuits have laid before the public the
inventories of their possessions and those plain and undisguised
statements could easily be found if there was any sincere desire to
get at the truth. Thus Foley has published in his "Records of the
English Province" (Introd., 139) an exact statement of the annual
revenues of the various houses for one hundred and twenty years.
Dühr in the "Jesuiten-fabeln" (606 sqq.) gives many figures of the
same kind for Germany. Indeed the Society has been busy from the
beginning trying to lay this financial ghost. Thus a demand for the
books was made as early as 1594 by Antoine Arnauld who
maintained that the French Jesuits enjoyed an annual revenue of
1,200,000 livres, which in our day would amount to $1,800,000.
Possibly some of the reverend Fathers nourished the hope that he
might be half right, but an official scrutiny of the accounts revealed
the sad fact that their twenty-five colleges and churches with a staff
of from 400 to 500 persons could only draw on 60,000 livres; which
meant at our values $90,000 a year — a lamentably inadequate
capital for the gigantic work which had been undertaken. Arnaulds
under different names have been appearing ever since.
How this "vast wealth" is accumulated, might also possibly be
learned by a visit to the dwelling-quarters of any Jesuit
establishment, so as to see at close range the method of its
domestic economy. Every member of the Society, no matter how
distinguished he is or may have been, occupies a very small,
uncarpeted room whose only furniture is a desk, a bed, a wash-
stand, a clothes-press, a prie-dieu, and a couple of chairs. On the
whitewashed wall there is probably a cheap print of a pious picture
which suggests rather than inspires devotion. This room has to be
swept and cared for by the occupant, even when he is advanced in
age or has been conspicuous in the Society, "unless for health's sake
or for reasons of greater moment he may need help." The clothing
each one wears is cheap and sometimes does service for years;

there is a common table; no one has any money of his own, and he
has to ask even for carfare if he needs it. If he falls sick he is
generally sent to an hospital where, according to present
arrangements, the sisters nurse him for charity, and he is buried in
the cheapest of coffins, and an inexpensive slab is placed over his
remains.
Now it happens that this method of living admits of an enormous
saving, and it explains how the 17,000 Jesuits who are at present in
the Society are able not only to build splendid establishments for
outside students, but to support a vast number of young men of the
Order who are pursuing their studies of literature, science,
philosophy, and theology, and who are consequently bringing in
nothing whatever to the Society for a period of eleven years, during
which time they are clothed, fed, cared for when sick, given the use
of magnificent libraries, scientific apparatus, the help of
distinguished professors, travel, and even the luxuries of villas in the
mountains or by the sea during the heats of summer. It will,
perhaps, be a cause of astonishment to many people to hear that
this particular section of the Order, thanks to common life and
economic arrangements, could be maintained year after year when
conditions were normal at the amazingly small outlay of $300 or
$400 a man. Of course, some of the Jesuit houses have been
founded, and devoted friends have frequently come to their rescue
by generous donations, but it is on record that in the famous royal
foundation of La Flèche, established by Henry IV, where one would
have expected to find plenty of money, the Fathers who were
making a reputation in France by their ability as professors and
preachers and scientific men were often compelled to borrow each
other's coats to go out in public. Such is the source of Jesuit wealth.
"They coin their blood for drachmas."
Failing to explain the Jesuits' pedagogical success by their
wealth, it has been suggested that their popularity in the
seventeenth and eighteenth centuries arose from the fact that it was
considered to be "good form" to send one's boys to schools which
were frequented by princes and nobles; but that would not explain

how they were, relatively, just as much favored in India and Peru as
in Germany or France. Indeed there was an intense opposition to
them in France, particularly on the part of the great educational
centres of the country, the universities: first, because the Jesuits
gave their services for nothing, and secondly because the teaching
was better, but chiefly, according to Boissier, who cites the authority
of three distinguished German pedagogues of the sixteenth century
— Baduel, Sturm, and Cordier — "because to the disorder of the
university they opposed the discipline of their colleges, and at the
end of three or four years of higher studies, regularly graduated
classes of upright, well-trained men." (Revue des Deux Mondes,
Dec., 1882, pp. 596, 610).
Compayré, who once figured extensively in the field of
pedagogical literature, finds this moral control an objection. He says
it was making education subsidiary to a "religious propaganda." If
this implies that the Society considers that the supreme object of
education is to make good Christian men out of their pupils, it
accepts the reproach with pleasure; and, there is not a Jesuit in the
world who would not walk out of his class to-morrow, if he were told
that he had nothing to do with the spiritual formation of those
committed to his charge. Assuredly, to ask a young man in all the
ardor of his youth to sacrifice every worldly ambition and happiness
to devote himself to teaching boys grammar and mathematics, to be
with them in their sports, to watch over them in their sleep, to be
annoyed by their thoughtlessness and unwillingness to learn; to be,
in a word, their servant at every hour of the day and night, for
years, is not calculated to inflame the heart with enthusiasm. The
Society knows human nature better, and from the beginning, its only
object has been to develop a strong Christian spirit in its pupils and
to fit them for their various positions in life. It is precisely because of
this motive that it has incurred so much hatred, and there can be no
doubt that if it relinquished this object in its schools, it would
immediately enjoy a perfect peace in every part of the world.
Nor can their educational method be charged with being an
insinuating despotism, as Compayré insists, which robs the student

of the most precious thing in life, personal liberty; nor, as Herr
describes it, "a sweet enthrallment and a deformation of character
by an unfelt and continuous pressure" (Revue universitaire, I, 312).
"The Jesuit," he says, "teaches his pupils only one thing, namely to
obey," which we are told, "is, as M. Aulard profoundly remarks, the
same thing as to please" (Enquête sur l'enseignement secondaire, I,
460). In the hands of the Jesuit, Gabriel Hanotaux tells us, the child
soon becomes a mechanism, an automaton, apt for many things,
well-informed, polite, self-restrained, brilliant, a doctor at fifteen,
and a fool ever after. They become excellent children, delightful
children, who think well, obey well, recite well, and dance well, but
they remain children all their lives. Two centuries of scholars were
taught by the Jesuits, and learned the lessons of Jesuits, the
morality of the Jesuits, and that explains the decadence of character
after the great sixteenth century. If there had not been something in
our human nature, a singular resource and things that can not be
killed, it was all up with France, where the Order was especially
prosperous.
As an offset to this ridiculous charge, the names of a few of "this
army of incompetents," these men marked by "decadence of
character," might be cited. On the registers of Jesuit schools are the
names of Popes, Cardinals, bishops, soldiers, magistrates,
statesmen, jurists, philosophers, theologians, poets and saints. Thus
we have Popes Gregory XIII, Benedict XIV, Pius VII, Leo XIII, St.
Francis of Sales, Cardinal de Bérulle, Bossuet, Belzunce, Cardinal de
Fleury, Cardinal Frederico Borromeo, Fléchier, Cassini, Séquier,
Montesquieu, Malesherbes, Tasso, Galileo, Corneille, Descartes,
Molière, J. B. Rousseau, Goldoni, Tournefort, Fontenelle, Muratori,
Buffon, Gresset, Canova, Tilly, Wallenstein, Condé, the Emperors
Ferdinand and Maximilian, and many of the princes of Savoy,
Nemours and Bavaria. Even the American Revolutionary hero, Baron
Steuben, was a pupil of theirs in Prussia, and omitting many others,
nearly all the great men of the golden age of French literature
received their early training in the schools of the Jesuits.

It is usual when these illustrious names are referred to, for
someone to say: "Yes, but you educated Voltaire." The implied
reproach is quite unwarranted, for although François Arouet, later
known as Voltaire, was a pupil at Louis-le-Grand, his teachers were
not at all responsible for the attitude of mind which afterwards made
him so famous or infamous. That was the result of his home training
from his earliest infancy. In the first place, his mother was the
intimate friend of the shameless and scoffing courtesan of the
period, Ninon de l'Enclos, and his god-father was Chateauneuf, one
of the dissolute abbés of those days, whose only claim to their
ecclesiastical title was that, thanks to their family connections, they
were able to live on the revenues of some ecclesiastical
establishment. This disreputable god-father had the additional
distinction of being one of Ninoñ's numerous lovers. It was he who
had his fileul named in her will, and he deliberately and
systematically taught him to scoff at religion, long before the
unfortunate child entered the portals of Louis-le-Grand. Indeed,
Voltaire's mockery of the miracles of the Bible was nothing but a
reminiscence of the poem known as the "Moïsade" which had been
put in his hands by Chateauneuf and which he knew by heart. The
wonder is that the Jesuits kept the poor boy decent at all while he
was under their tutelage. Immorality and unbelief were in his home
training and blood.
Another objection frequently urged is that the Jesuits were really
incapable of teaching Latin, Greek, mathematics or philosophy, and
that in the last mentioned study they remorselessly crushed all
originality.
To prove the charge about Latin, Gazier, a doctor of the
Sorbonne, exhibited a "Conversation latine, par Mathurin Codier,
Jésuite." Unfortunately for the accuser, however, it was found out
that Codier not only was not a Jesuit, but was one of the first
Calvinists of France. Greek was taught in the lowest classes; and in
the earliest days the Society had eminent Hellenists who attracted
the attention of the learned world, such as: Gretser, Viger, Jouvancy,
Rapin, Brumoy, Grou, Fronton du Duc, Pétau, Sirmond, Garnier and

Labbe. The last mentioned was the author of eighty works and his
"Tirocinium linguæ græcæ" went through thirteen or fourteen
editions. At Louis-le-Grand there were verses and discourses in
Greek at the closing of the academic year. Bernis says he used to
dream in Greek. There were thirty-two editions of Gretser's
"Rudimenta linguæ græcæ," and seventy-five of his "Institutiones."
Huot, when very young, began a work on Origen, and Bossuet,
when still at college, became an excellent Greek scholar. They were
both Jesuit students.
"The Jesuits were also responsible for the collapse of scientific
studies," says Compayré (193,197). The answer to this calumny is
easily found in the "Monumenta pedagogica Societatis Jesu" (71-78),
which insists that "First of all, teachers of mathematics should be
chosen who are beyond the ordinary, and who are known for their
erudition and authority." This whole passage in the "Monumenta,"
was written by the celebrated Clavius. Surely it would be difficult to
get a man who knew more about mathematics than Clavius. It will
be sufficient to quote the words of Lalande, one of the greatest
astronomers of France, who, it may be noted incidentally, was a
pupil of the Jesuits. In 1800 he wrote as follows: "Among the most
absurd calumnies which the rage of Protestants and Jansenists
exhale against the Jesuits, I found that of La Chalotais, who carried
his ignorance and blindness to such a point as to say that the Jesuits
had never produced any mathematicians. I happened to be just then
writing my book on 'Astronomy,' and I had concluded my article on
'Jesuit Astronomers,' whose numbers astonished me. I took occasion
to see La Chalotais, at Saintes, on July 20, 1773, and reproached
him with his injustice, and he admitted it."
"As for history," says Compayré, "it was expressly enjoined by
the 'Ratio' that its teaching should be superficial." And his assertion,
because of his assumed authority, is generally accepted as true,
especially as he adduces the very text of the injunction which says:
"Historicus celerius excurrendus," namely "let historians be run
through more rapidly." Unfortunately, however, the direction did not
apply to the study of history at all, but to the study of Latin, and

meant that authors like Livy, Tacitus, and Cæsar were to be gone
through more expeditiously than the works of Cicero, for example,
who was to be studied chiefly for his exquisite style. In brief, the
charge has no other basis than a misreading, intentional or
otherwise, of a school regulation.
The same kind of tactics are employed to prove that no
philosophy was taught in those colleges, in spite of the fact that it
was a common thing for princes and nobles and statesmen to come
not only to listen to philosophical disputations in the colleges, in
which they themselves had been trained, but to take part in them.
That was one of Condé's pleasures; and the Intendant of Canada,
the illustrious Talon, was fond of urging his syllogisms against the
defenders in the philosophical tournaments of the little college of
Quebec. Nor were those pupils merely made to commit to memory
the farrago of nonsense which every foolish philosopher of every age
and country had uttered, as is now the method followed in non-
Catholic colleges. The Jesuit student is compelled not only to state
but to prove his thesis, to refute objections against it, to retort on
his opponents, to uncover sophisms and so on. In brief, philosophy
for him is not a matter of memory but of intelligence. As for
independence of thought, a glance at their history will show that
perhaps no religious teachers have been so frequently cited before
the Inquisition on that score, and none to whom so many theological
and philosophical errors have been imputed by their enemies, but
whose orthodoxy is their glory and consolation.
Their failure to produce anything in the way of painting or
sculpture has also afforded infinite amusement to the critics,
although it is like a charge against an Academy of Medicine for not
having produced any eminent lawyers, or vice versa. It is true that
Brother Seghers had something to do with his friend Rubens, and
that a Spanish coadjutor was a sculptor of distinction, and that a
third knew something about decorating churches, and that two were
painters in ordinary for the Emperor of China, but whose
masterpieces however have happily not been preserved. Hüber, an
unfriendly author, writing about the Jesuits, names Courtois, known

as Borgognone, by the Italians, who was a friend of Guido Reni;
Dandini, Latri, Valeriani d'Aquila and Castiglione, none of whom,
however, has ever been heard of by the average Jesuit. An eminent
scholar once suggested that possibly the elaborate churches of the
Compañía, which are found everywhere in the Spanish-American
possessions, may have been the work of the lay-brothers of the
Society. But a careful search in the menologies of the Spanish
assistancy has failed to reveal that such was the case. That,
however, may be a piece of good fortune, for otherwise the Society
might have to bear the responsibility of those overwrought
constructions, in addition to the burden which is on it already of
having perpetrated what is known as the "Jesuit Style" of
architecture. From the latter accusation, however, a distinguished
curator of the great New York Metropolitan Museum of Art, Sir
Caspar Purdon Clarke, in an address to an assembly of artists and
architects, completely exonerated the Society. "The Jesuit Style," he
said, "was in existence before their time, and," he was good enough
to add, "being gentlemen, they did not debase it, but on the
contrary elevated and ennobled it and made it worthy of artistic
consideration."
So, too, the Order has not been conspicuous for its poets. One
of them, however, Robert Southwell, was a martyr, and wore a
crown that was prized far more by his brethren than the laurels of a
bard. He was born at Norfolk on February 21, 1561, and entered the
Society at Rome in 1578. Singularly enough, the first verses that
bubbled up from his heart, at least of those that are known, were
evoked by his grief at not being admitted to the novitiate. He was
too young to be received, for he was only seventeen, and conditions
in England did not allow it; but his merit as a poet may be inferred
from an expression of Ben Jonson that he would have given many of
his works to have written Southwell's "Burning Babe," and, according
to the "Cambridge History of Literature" (IV, 129), "though Southwell
may never have read Shakespeare, it is certain that Shakespeare
read Southwell." Of course, his poems are not numerous, for though
he may have meditated on the Muse while he was hiding in out of

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