Central Limit Theorem for a Class of Random Measures Associated with Germ–Grain Models

Abstract

The germ-grain model is defined as the union of independent identically distributed compact random sets (grains) shifted by points (germs) of a point process. The paper introduces a family of stationary random measures in Rd generated by germ-grain models and defined by the sum of contributions of non-overlapping parts of the individual grains. The main result of the paper is the central limit theorem for these random measures, which holds for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. It is shown that this construction of random measures includes those random measures obtained by positively extended intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

Full text

Central limit theorem for a class of random
measures associated with germ-grain models
LOTHAR HEINRICH, University of Augsburg 1
ILYA S. MOLCHANOV, University of Glasgow 2
Abstract
The germ-grain model is defined as the union of independent identically
distributed compact random sets (grains) shifted by points (germs) of a point
process. The paper introduces a family of stationary random measures in
Rdgenerated by germ-grain models and defined by the sum of contributions
of non-overlapping parts of the individual grains. The main result of the
paper is the central limit theorem for these random measures, which holds
for rather general independently marked germ-grain models, including those
with non-Poisson distribution of germs and non-convex grains. It is shown
that this construction of random measures includes those random measures
obtained by positively extended intrinsic volumes. In the Poisson case it is
possible to prove a central limit theorem under weaker assumptions by using
approximations by m-dependent random fields. Applications to statistics of
the Boolean model are also discussed. They include a standard way to derive
limit theorems for estimators of the model parameters.
β-mixing, boolean model, germ-grain model, intrinsic volumes,
m-dependent random field, random measure, random set, weak
dependence
AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60D05;
SECONDARY 52A22, 60F05, 60G57.
1Postal address: Institut f¨ur Mathematik, Universit¨at Augsburg, D-86135 Augsburg, Germany.
2Postal address: University of Glasgow, Department of Statistics, Glasgow G12 8QW, Scotland, U.K.

1 Introduction
The ergodic theorem for spatial processes [23] establishes the existence of spatial averages
when a sampling window expands unboundedly in all directions. In many applications
such averages can be interpreted as intensities of stationary random measures concentrated
on random closed sets in Rd. For example, a point process corresponds to a random
counting measure, a random closed set with non-empty interior gives rise to the random
volume measure. Further examples are provided by geometric measures, e.g., surface
and curvature measures of random sets. The relationships between intensities of such
measures and parameters of the underlying random set or point process are the basis of
the method of moments in statistics of stationary random sets [21, 27, 31].
A simple example states that, for any ergodic random closed set Zin Rd, the volume
fraction inside window W(the part of the window covered), converges almost surely
to its expectation called the volume fraction of Z, while Wis assumed to expand to
the whole space in a ‘regular way’ [4, p. 332]. The corresponding central limit theorem
was established in [1, 17, 18]. In fact, it is related to weak dependence properties of
the indicator random field ζ(x) = 1Z(x) and limit theorems for additive functionals
of random fields. Similar results are known for counting measures generated by point
processes [14, 15].
For more general functionals (which include, e.g., the boundary length or surface area)
the convergence of spatial averages was considered in [23] and [33]. However, until now
a general approach to derive limit theorems for such functionals seems to be unknown.
It should be noted that these limit theorems are very important for the construction of
confidence intervals for estimators in statistics of stationary random sets. Variances for
functionals related to surface and fibre processes has been found in [2, 3]. However, in
the latter case the whole situation resembles ones with marked point processes, since the
individual fibres are directly observable, see [27, Chapter 9].
Central imit theorems for geometric functionals other than volume fractions and the
number of points were proved in [22] and [20] for the convexity number and the surface
measure of the Boolean model Z. In this paper we consider more general models of random
sets and geometric functionals and prove the corresponding central limit theorem. Also,
we do not need the boundedness of the grain assumed in [20] for the case of surface
measures.
It is known that ergodicity and first moment assumptions are not sufficient to prove
a central limit theorem. One has to impose further conditions on both the random sets
and the corresponding random measures. In the present paper we are interested in a
class of random measures associated with independently marked germ-grain models. To
begin with, we recall the definition of germ-grain models and introduce a class of random
measures associated with them (Section 3). The germ-grain model is composed from a
2

sequence of independent identically distributed random sets (grains) shifted by the points
(germs) of a point process. Germ-grain models with compact grains are widely used in
stochastic geometry and generalise the well-known Boolean model, which appears if the
point process of germs is Poisson. Roughly speaking, these random measures associated
with germ-grain models are defined through sums of contributions of the exposed (not
covered by all other grains) parts of the individual grains. In the simplest case, this
construction can be used to obtain the surface area of the germ-grain model.
Statistical properties of the random measures introduced here are studied in Sections 4
and 5. In particular, higher-order mixed moment measures are given in Section 4 and
the absolute regularity (or β-mixing) is proved in Section 5. We show that the β-mixing
property of the germ process implies β-mixing of the germ-grain model and the associated
random measure provided the diameter of the typical grain has sufficiently high moments.
Section 6 establishes the central limit theorem for the associated random measures. Sec-
tion 7 deals with the Boolean model for which the central limit theorem holds under very
mild (in fact, optimal) assumptions.
It is shown in Appendix that the random measures introduced here yield, in particular,
the positive extensions of the intrinsic volumes defined by Matheron [19, Section 4.7].
Mean value formulae for extended intrinsic volumes are very important in statistics of
the Boolean model. Therefore, the general results derived in this paper give rise to
new mean-value formulae and provide a unified approach to limit theorems for spatial
averages. A number of examples are considered in Section 8. Section 9 outlines several
possible generalisations.
2 List of Basic Notation
[Rd,Bd]:d-dimensional Euclidean space with Borel σ-algebra;
Bd
0: family of bounded Borel sets in Rd;
kxk: Euclidean norm of x∈Rd;
kKk= sup{kxk:x∈K},K∈Bd
0;
|B|:d-dimensional Lebesgue measure (or volume) of B∈Bd
0;
Br(x) = {y∈Rd:kx−yk ≤ r}for x∈Rdand r > 0;
1B(x) = 1 if x∈Band = 0 otherwise (indicator function of B);
Fc: complement of set Fin the underlying space;
∂F : boundary of F⊂Rd;
[F, σf]: family of closed sets in Rdwith the σ-algebra generated by the hit-or-miss
topology, see [19, 27];
3

[K, σk] : family of compact sets in Rdwith σ-algebra σk={A ∩ K :A ∈ σf};
ˇ
K={−x:x∈K}for K∈ K;
⊕: Minkowski addition, i.e., F1⊕F2={x+y:x∈F1, y ∈F2},F1, F2⊂Rd;
C: family of convex compact sets in Rd;
[Ω,A,P] : hypothetical common probability space on which all occurring random ele-
ments are defined;
E: expectation with respect to P;
δx: Dirac measure concentrated at point x.
3 Germ-Grain Models and Associated Random
Measures
Stationary random closed set Zis a random element in space [F, σf] such that Zhas
the same distribution as Z+xfor all x∈Rd, see [19, 27]. The most important model
of stationary random sets is the germ-grain model. This model is defined by means of a
stationary marked point process
Ψm=X
i:i≥1
δ[Xi;Zi]
in Rdwith the mark space [K, σk] , where the stationarity is understood with respect to its
first component, see [27] for further details on marked point processes. The corresponding
germ-grain model Zis the set-theoretic union
Z=[
i:i≥1
(Xi+Zi).(3.1)
The points Xiare called germs, while the sets Ziare called grains, see [8, 27]. The model
exists as soon as Zis closed and Z6=Rd, see [11].
Throughout we assume that the corresponding unmarked point process
Ψ(·) = Ψm((·)× K) = X
i:i≥1
δXi
is simple and has a positive and finite intensity λ=EΨ([0,1]d). Remember that Ψ is
a random element in the space Nof locally finite counting measures ψon Rd, where N
is equipped with the σ-algebra Ngenerated by sets of the form {ψ∈N:ψ(B) = k}
for B∈Bd
0and k= 0,1, . . . . The distribution of Ψ is denoted by P, so that P(Y) =
P{Ψ∈Y}for each Y∈N.
4

In this paper we will consider only independently marked point processes, where
{Zi, i ≥1}is a sequence of i.i.d. copies of a random compact set Z0being indepen-
dent of Ψ. The random compact set Z0is called the typical grain. Note that Z0is a
random element in the space [K, σk] with distribution denoted by Q. Then random set
(3.1) is closed and different from Rd(so the germ-grain model exists) if
E|Z0⊕Br(0)|<∞for some r > 0 (3.2)
(e.g., if EkZ0kd<∞), see [11]. If the point process of germs Ψ is Poisson, Zis said to
be a Boolean model [19, 27].
Germ-grain models give rise to a number of random measures on Bd
0, see [19]. In the
simplest case it is possible to define a random measure ηby η(W) = |Z∩W|,W∈Bd
0.
Another random measure can be defined by taking the surface area of Zinside W. Further
random measures are the Minkowski random measures introduced in [19]. It is worthwhile
to note that many interesting measures associated with Zcan be decomposed into the sum
with respect to all individual grains. In this sum each grain contributes to the resulting
measure with some (possibly random) weight. For further information on general random
measures the reader is referred to [16], [4, Chapter 6] and [27].
Below we present a unified approach to random measures associated with germ-grain
models. Let [U,U] be a compact metric space with the corresponding Borel σ-algebra (in
many examples Uis the unit sphere in Rd). Furthermore, let [Σ,B(Σ)] be the product
space U×Rdwith the corresponding product σ-algebra. By M(Σ) we denote the family
of locally finite measures on Σ equipped with the σ-algebra M(Σ) generated by sets of
the form {µ∈M(Σ) : a≤µ(S)≤b}for S∈B(Σ) and 0 ≤a < b < ∞.
Let
H•:K 7→ M(Σ)
be a (σk,M(Σ))-measurable mapping. For each K∈ K ,HK(·) is a finite measure on
B(Σ) . For the sake of convenience we write HK(Γ, W ) instead of HK(Γ ×W) for Γ ∈U
and W∈Bd
0. The measure-valued function His the basic object to construct random
measures associated with the germ-grain model.
We assume that the map Hsatisfies the following conditions
HK(Γ, W ) = HK(Γ, W ∩K),(3.3)
and
HK+x(Γ, W +x) = HK(Γ, W ) (3.4)
for all K∈ K , Γ ∈U,W∈Bd
0, and x∈Rd.
Moreover, we assume that
EHZ0(U,Rd)<∞,(3.5)
5

so that
H(Γ, W ) = EHZ0(Γ, W ) (3.6)
is a finite deterministic measure on B(Σ) .
Remark 3.1.It is also possible to consider signed measures H. Then the finiteness con-
dition must be replaced by the finiteness of the expected total variation of HZ0.
We are now in a position to define a locally finite random measure η(Γ, W ) on B(Σ)
by
η(Γ, W ) = X
i:i≥1
HXi+Zi(Γ, W \Ξi),Γ∈U, W ∈Bd
0,(3.7)
where
Ξi=[
j:i6=i
(Xj+Zj), i ≥1.(3.8)
Roughly speaking, HXi+Zi(Γ, W \Ξi) is a contribution of the exposed part of the ith grain
inside the window Wand satisfying some conditions specified by Γ ⊆U, see Figure 1. Figure 1
hereThe local finiteness of ηresults from conditions which ensure the existence of the
germ-grain model, for example, (3.2) is sufficient. Obviously, η(Γ, W +x) equals η(Γ, W )
in distribution for all Γ ∈U,W∈Bd
0, and x∈Rd, i.e., η(Γ,·) is a stationary random
measure on Bd
0.
To avoid confusion, we put together our basic assumptions.
Basic assumptions:
(i) Ψmis a simple stationary independently marked point process. The corresponding
unmarked point process Ψ has a finite positive intensity λ.
(ii) The measurable mapping K7→ HK(·) satisfies (3.3) and (3.4).
(iii) The typical grain Z0satisfies (3.2) and (3.5).
Consider family of measures HK(·) which admit integral representation
HK(Γ, W ) = Z
Γ
1W(ℓ(u, K))ΥK(du), K ∈ K ,(3.9)
where Υ•:K 7→ M(U) is a (σk,M(U))-measurable mapping. Here M(U) is the set
of finite measures on [U,U] , and M(U) is the σ-algebra generated by sets of the form
{µ∈M(U) : a≤µ(U)≤b}for U∈Uand 0 ≤a < b < ∞. Assume that ΥK(·) is
invariant under translations of K, i.e., ΥK(·) = ΥK+x(·) , and the mapping ℓ:U×K 7→ Rd
is (U⊗σk,Bd)-measurable and satisfies the conditions
ℓ(u, K)∈Kand ℓ(u, K +x) = ℓ(u, K) + x
6

for all K∈ K ,u∈Uand x∈Rd. Then both (3.3) and (3.4) hold; HK(Γ,Rd) = ΥK(Γ)
for all K∈ K; and (3.7) can be written as
η(Γ, W ) = X
i:i≥1Z
Γ
1W(Xi+ℓ(u, Zi)) Y
j:j6=i1−1Xj+Zj(Xi+ℓ(u, Zi))ΥZi(du).(3.10)
In this case the following useful lemma holds.
Lemma 3.1. If HK(·)admits integral representation (3.9), then, for each measurable
function f:Rd7→ [0,∞),
EZ
W
f(y)HZ0(Γ, dy) = Z
W
f(y)H(Γ, dy) = Z
KZ
Γ
1W(ℓ(u, K))f(ℓ(u, K))ΥK(du)Q(dK).
4 Moment Measures of η
Below we will find the moment measures of the random measure ηgiven by (3.7). Fur-
ther P!
x1,...,xk(·) denotes the kth-order (reduced) Palm distribution of Ψ with respect to
x1,...,xk∈Rd(see [16] and [27, pp. 121–124]), and
G!
x1,...,xk[f] = Z
NY
zi∈ψ
f(zi)P!
x1,...,xk(dψ)
is the probability generating functional with respect to the Palm distribution P!
x1,...,xk.
For each compact set Kdenote
τK(z) = Pz /∈ˇ
Z0⊕K=P{Z0∩(K−z) = ∅} ,(4.1)
and write shortly τyinstead of τ{y}.
The following lemma gives the first moment measure of η.
Lemma 4.1. Under the basic assumptions,
Eη(Γ, W ) = λ(Γ)|W|,(4.2)
where
λ(Γ) = λZ
Rd
G!
0[τy]H(Γ, dy).(4.3)
In particular, if Ψis a stationary Poisson process with intensity λ, then
λ(Γ) = λexp{−λE|Z0|}H(Γ,Rd) = λ(1 −p)H(Γ,Rd),(4.4)
where p=P{0∈Z}is the volume fraction of Z.
7

Proof. Note that
HXi+Zi(Γ, W ) = HZi(Γ, W −Xi) = Z
Rd
1W(Xi+y)HZi(Γ, dy).
Using the refined Campbell theorem [4, p. 116] and the independent marking we get
Eη(Γ, W ) = EX
i:i≥1Z
Rd
1W(Xi+y)Y
j:j6=i1−1Xj+Zj(Xi+y)HZi(Γ, dy)
=Z
NX
i:xi∈ψZ
KZ
Rd
1W(xi+y)Y
j:xj∈ψ−δxi
E1−1xj+Z0(xi+y)HK(Γ, dy)Q(dK)P(dψ)
=λZ
RdZ
KZ
Rd
1W(x+y)Z
NY
j:xj∈ψ
E1−1ˇ
Z0(xj−x−y)P!
x(dψ)HK(Γ, dy)Q(dK)dx
=λZ
RdZ
KZ
Rd
1W(x+y)Z
NY
j:xj∈ψ
E1−1ˇ
Z0(xj−y)P!
0(dψ)HK(Γ, dy)Q(dK)dx .
By definition of the probability generating functional G!
0with respect to the Palm distri-
bution P!
0,
Z
NY
j:xj∈ψ
E1−1ˇ
Z0(xj−y)P!
0(dψ) = Z
NY
j:xj∈ψ
Pxj/∈ˇ
Z0+yP!
0(dψ) = G!
0[τy].
Now (4.2) and (4.3) are easy to derive, since
Eη(Γ, W ) = λ|W|Z
KZ
Rd
G!
0[τy]HK(Γ, dy)Q(dK) = λ|W|Z
Rd
G!
0[τy]H(Γ, dy).
If Ψ is a Poisson process, then (4.4) follows from Slivnyak’s theorem [4, p. 459] and the
explicit formula for the probability generating functional [4, p. 225].
Remark 4.1.If Hadmits integral representation (3.9), then, by Lemma 3.1,
λ(Γ) = λZ
KZ
Γ
G!
0τℓ(u,K)ΥK(du)Q(dK ).(4.5)
We proceed with the following general result which gives higher-order mixed moment
measures of η. Note that the kth-order factorial moment measure of Ψ is defined by
α(k)(B1× · · · × Bk) = EX∗
x1,...,xk∈Ψ
1B1(x1)···1Bk(xk),
where P∗
x1,...,xk∈ψdesignates the sum over all k-tuples of pairwise distinct atoms of ψ∈N.
8

Lemma 4.2. In addition to the above basic assumptions let the kth-order factorial mo-
ment measure α(k)of Ψexist, EHk
Z0(U,Rd)<∞, and, for some r > 0,
Z
(Rd)ν
ν
Y
j=1
EhHkj
Z0(U, Br(−xj))iα(ν)(d(x1,...,xk)) <∞(4.6)
for all 1≤ν≤kand positive numbers k1,...,kνsatisfying k1+· · · +kν=k.
Then, for every Γ1,...,Γk∈Uand W1,...,Wk∈Bd
0,
Eη(Γ1, W1)···η(Γk, Wk)=
k
X
ν=1 X
I1∪···∪Iν={1,...,k}Z
(Rd)νZ
(K)νZ
(Rd)k
ν
Y
j=1 Y
ij∈Ij
1Wij(xj+yij)
×
ν
Y
i,j=1
i6=jY
ij∈Ij1−1ˇ
Ki+yij(xi−xj)G!
x1,...,xνhτ{xj+yij: 1≤j≤ν, ij∈Ij}i
×
ν
Y
j=1 Y
ij∈Ij
HKj(Γij, dyij)
ν
Y
j=1
Q(dKj)α(ν)(d(x1,...,xν)) .(4.7)
Here the sum PI1∪···∪Iν={1,...,k}is taken over all non-empty partitions of the set {1,...,k}
into ν∈ {1,...,k}subsets I1,...,Iν.
Remark 4.2.If Ψ is a stationary Poisson process, then α(ν)(d(x1,...,xν)) = λνdx1...dxν,
condition (4.6) follows from EHk
Z0(U,Rd)<∞, and
G!
x1,...,xνhτ{xj+yij: 1≤j≤ν, ij∈Ij}i= exp 


−λE
ν
[
j=1 [
ij∈Ijˇ
Z0+xj+yij



.
Proof. For any k-tuple of functions fj:Rd7→ R1,j= 1,...,k, with bounded supports
the identity
X
x1,...,xk∈ψ
f1(x1)···fk(xk) =
k
X
ν=1 X
I1∪···∪Iν={1,...,k}X∗
y1,...,yν∈ψ
ν
Y
j=1 Y
ij∈Ij
fij(yj)
enables us to write
Eη(Γ1, W1)···η(Γk, Wk)
=
k
X
ν=1 X
I1∪···∪Iν={1,...,k}
E"Z
(Rd)kX∗
q1,...,qν≥1
ν
Y
j=1 Y
ij∈Ij
1Wij(Xqj+yij)
×
ν
Y
j=1 Y
pj≥1
pj6=qjY
ij∈Ij1−1Zpj−yij(Xqj−Xpj)ν
Y
j=1 Y
ij∈Ij
HZqj(Γ, dyij)#.
9

By simple manipulations it is seen that, for fixed q1,...,qνand I1,...,Iν, the product
ν
Y
j=1 Y
pj≥1
pj6=qjY
ij∈Ij1−1Zpj−yij(Xqj−Xpj)=
ν
Y
j=1 Y
pj≥1
pj6=qjY
ij∈Ij1−1ˇ
Zpj+yij(Xpj−Xqj)
equals
ν
Y
j=1
ν
Y
j=1
j6=iY
ij∈Ij1−1ˇ
Zqi+yij(Xqi−Xqj)Y
p≥1
p6=q1,...,qν
ν
Y
j=1 Y
ij∈Ij1−1ˇ
Zp+yij(Xp−Xqj).
Exploiting the independence of marks associated with distinct atoms and the indepen-
dence between the marks and the point process Ψ combined with a multiple use of Fubini’s
theorem we find that
E"X∗
q1,...,qν≥1Z
(Rd)k
ν
Y
j=1 Y
ij∈Ij
1Wij(Xqj+yij)
ν
Y
i,j=1
i6=jY
ij∈Ij1−1ˇ
Zqi+yij(Xqi−Xqj)
×Y
p≥1
p6=q1,...,qν
ν
Y
j=1 Y
ij∈Ij1−1ˇ
Zp+yij(Xp−Xqj)ν
Y
j=1 Y
ij∈Ij
HZqj(Γ, dyij)#
=Z
(Rd)ν
E"Z
(Rd)k
ν
Y
j=1 Y
ij∈Ij
1Wij(xj+yij)
ν
Y
i,j=1
i6=jY
ij∈Ij1−1ˇ
Zi+yij(xi−xj)#
×Z
NY
y∈ψ
E

ν
Y
j=1 Y
ij∈Ij1−1ˇ
Z0+yij(y−xj)
P!
x1,...,xν(dψ)E

ν
Y
j=1 Y
ij∈Ij
HZj(Γ, dyij)

×α(ν)(d(x1,...,xν)) .
Here we have also used the relationship
Z
NX∗
x1,...,xν∈ψ
f(x1,...,xν, ψ −δx1−...−δxν)P(dψ)
=Z
(Rd)νZ
N
f(x1,...,xν, ψ)P!
x1,...,xν(dψ)α(ν)(d(x1,...,xν)) ,
which holds for any (Bd)ν⊗N-measurable function f: (Rd)ν×N7→ R1.
Finally, since
E

ν
Y
j=1 Y
ij∈Ij1−1ˇ
Z0+yij(y−xj)
=P


y /∈
ν
[
j=1 [
ij∈Ij
(ˇ
Z0+yij+xj)


,
10

the definition of the probability generating functional G!
x1,...,xντ{xj+yij: 1≤j≤ν, ij∈Ij}yields
(4.7). The liberal use of Fubini’s theorem is justified because the right-hand side of (4.7)
is bounded by
k
X
ν=1 X
k1+···+kν=k
ki≥1
k!
k1!···kν!Z
(Rd)ν
ν
Y
j=1
EhHkj
Z0(U, Br(−xj))iα(ν)(d(x1,...,xν)) ,
where r > 0 is chosen such that W1∪ · · · ∪ Wk⊆Br(0) . Now (4.6) implies that this sum
is finite.
If the grain Z0is Hausdorff rectifiable (Hm-rectifiable) [35] with m < d , then Z
represents a particular class of the random processes of Hausdorff rectifiable closed sets
in Rd. This concept includes the well-known fibre and surface processes studied in [27].
Lemma 4.2 can be used to express the corresponding moment measures found in [35].
The second-order moment measures of ηdepend on the second-order moment measure
H(Γ1, W1; Γ2, W2) = E[HZ0(Γ1, W1)HZ0(Γ2, W2)] , W1, W2∈Bd
0,(4.8)
of HZ0(·) , and the measure
Φ(x; Γ1, W1; Γ2, W2) = EhHZ1(Γ1,(Zc
2+x)∩W1)HZ2(Γ2,(Zc
1−x)∩W2)i(4.9)
defined for two independent grains Z1and Z2having the same distribution as Z0. Fur-
thermore, α(2)
red (resp. γ(2)
red) is the reduced second-order factorial moment (resp. reduced
covariance) measure of Ψ , i.e., α(2)
red(B) = RNψ(B)P!
0(dψ) and γ(2)
red(B) = α(2)
red(B)−λ|B|
for B∈Bd
0.
Corollary 4.3. If Ψis a second-order point process, EH2
Z0(U,Rd)<∞and
Z
(Rd)2
H(U, W −x1)H(U, W −x2)α(2)(d(x1, x2)) <∞,
then
Eη(Γ1, W )η(Γ2, W )=λZ
(Rd)2
γW(y2−y1)G!
0τ{y1,y2}H(Γ1, dy1; Γ2, dy2)
+λZ
RdZ
(Rd)2
γW(x+y2−y1)G!
0,x τ{y1,x+y2}Φ(x; Γ1, dy1; Γ2, dy2)α(2)
red(dx),
where γW(x) = |W∩(W+x)|is the set-covariance function of W.
11

5 Absolute Regularity of the Random Measure η(Γ,·)
The absolute regularity coefficient (or β-mixing , or weak Bernoulli coefficient)β(X,Y)
between any two sub-σ-algebras X,Y⊂Ais defined by
β(X,Y) = 1
2sup X
kX
l
|P(Ak∩Bl)−P(Ak)P(Bl)|,(5.1)
see [29]. The supremum is taken over all pairs of finite partitions {Ak}and {Bl}of Ω
such that Ak∈Xand Bl∈Y. Standard measure-theoretic arguments ensure that the
supremum in (5.1) does not change its value if the Ak’s and Bl’s belong to semi-algebras
˜
Xand ˜
Ygenerating Xand Yrespectively.
To be general enough, we consider the random measure
η(f, ·) = X
i:i≥1Z
U
f(u)HZi(du, (·)\Ξi) (5.2)
for any measurable, bounded function f:U7→ [0,∞) . For any B∈Bddefine sub-σ-
algebras Aη(f)(B) (respectively AΨ(B)) generated by events {η(f, B′)∈[a, b)}(respec-
tively {Ψ(B′) = k}) for B′∈Bd
0,B′⊆B, Γ ∈U, 0 ≤a < b < ∞, and k= 0,1,2,....
Furthermore, define the restricted germ-grain model
Z(B) = [
i:Xi∈B
(Xi+Zi).
The following lemma will serve as a cornerstone for the later proof of the asymptotic
normality of η(f, W ) .
Lemma 5.1. For any two pairs of bounded Borel sets F, G and ˜
F , ˜
Gsuch that F⊆˜
F,
G⊆˜
Gand ˜
F∩˜
G=∅we have
β(Aη(f)(F),Aη(f)(G)) ≤β(AΨ(˜
F),AΨ(˜
G)) (5.3)
+ 2PnZ(˜
Fc)∩F6=∅o+ 2PnZ(˜
Gc)∩G6=∅o.
Remark 5.1.Estimate (5.3) is in the spirit of Mase [17] who derived upper bounds of the
α-mixing coefficient of germ-grain models. Explicit estimates of the β-mixing coefficient
of Voronoi tessellations and Poisson cluster processes are given in [13].
Proof. Let F1,...,Fp(resp. G1,...,Gq) form an arbitrary partition of F(resp. G).
Define the following dissections of the probability space Ω into disjoint events:
A¯
k=nη(f, F1)∈I(1)
k1,...,η(f, Fp)∈I(p)
kpo,
B¯
l=nη(f, G1)∈J(1)
l1,...,η(f, Gq)∈J(q)
lqo
12

for ¯
k= (k1,...,kp) and¯
l= (l1,...,lq) with ki= 1,...,mi,i= 1,...,p, and lj= 1,...,nj,
j= 1,...,q. The intervals I(i)
ki, ki= 1,...,mi,(resp. J(j)
lj, lj= 1,...,nj) are pairwise
disjoint and
I(1)
1∪ · · · ∪ I(i)
mi= [0,∞) (resp. J(1)
1∪ · · · ∪ J(j)
nj= [0,∞))
for any i= 1,...,p (resp. j= 1,...,q). According to the definition of the absolute
regularity coefficient β(Aη(f)(F),Aη(f)(G)) we have
β(Aη(f)(F),Aη(f)(G)) = 1
2sup X
¯
kX
¯
l
|P(A¯
k∩B¯
l)−P(A¯
k)P(B¯
l)|,
where supremum ranges over all partitions F1,...,Fp,G1,...,Gq,I(i)
1,...,I(i)
mifor i=
1,...,p and J(j)
1,...,J(j)
njfor j= 1,...,q. We compare the events A¯
kwith the corre-
sponding events ˜
A¯
k={η˜
F(f, F1)∈I(1)
k1,...,η˜
F(f, Fp)∈I(p)
kp}arising from the ‘truncated’
random measure η˜
F(f, ·) given by
η˜
F(f, ·) = X
i:i≥1
1˜
F(Xi)Z
U
f(u)HZi(du, (·)\Ξi).
Note that η(f, F ′) = η˜
F(f, F ′) for all F′⊆F, as soon as Z(˜
Fc)∩F=∅. Then (3.3)
yields
{Z(˜
Fc)∩F=∅} ∩ ˜
A¯
k⊆A¯
kand {Z(˜
Fc)∩F=∅} ∩ A¯
k⊆˜
A¯
k.
Therefore,
A¯
k△˜
A¯
k⊆ {Z(˜
Fc)∩F6=∅} ∩ A¯
k∪˜
A¯
k,
where A△B= (A∩Bc)∪(Ac∩B) designates the symmetric difference. Thus,
X
¯
k
P(A¯
k△˜
A¯
k)≤2PnZ(˜
Fc)∩F6=∅o.
Analogously,
X
¯
l
P(B¯
l△˜
B¯
l)≤2PnZ(˜
Gc)∩G6=∅o
for the events ˜
B¯
l={η˜
G(f, G1)∈J(1)
l1,...,η˜
G(f, Gq)∈J(q)
lq}.After elementary manipula-
tions we get
X
¯
kX
¯
l
|P(A¯
k∩B¯
l)−P(A¯
k)P(B¯
l)−(P(˜
A¯
k∩˜
B¯
l)−P(˜
A¯
k)P(˜
B¯
l))|
≤2X
¯
k
P(A¯
k△˜
A¯
k) + 2 X
¯
l
P(B¯
l△˜
B¯
l) (5.4)
≤4PnZ(˜
Fc)∩F6=∅o+ 4PnZ(˜
Gc)∩G6=∅o.
13

Note that ˜
A¯
kand ˜
B¯
lare conditionally independent given Ψ. Using arguments similar
to [34] and the formula (4.6) of [13] we obtain that
X
¯
kX
¯
l
|P(˜
A¯
k∩˜
B¯
l)−P(˜
A¯
k)P(˜
B¯
l)| ≤ 2β(AΨ(˜
F),AΨ(˜
G)) .
The latter estimate combined with (5.4) completes the proof of Lemma 5.1.
Next we specify (5.3) for
F= [−a, a]d, G =Rd\[−b, b]d,
˜
F= [−(a+ ∆), a + ∆]d,˜
G=Rd\[−(b−∆), b −∆]d
with 0 < a < b < ∞and ∆ = (b−a)/4 . Note that by a simple approximation argument
inequality (5.3) remains valid for unbounded Gand ˜
G.
Lemma 5.2. Let Ψbe a stationary point process with intensity λ > 0. Assume that
EkZ0kd<∞.Then, for the above defined F, ˜
Fand G, ˜
G,
PnZ(˜
Fc)∩F6=∅o≤λd2d1 + a
∆d−1
∞
Z
∆
xddD(x)
and
PnZ(˜
Gc)∩G6=∅o≤λd2d1 + 3a
∆d−1∞
Z
∆
xddD(x),
where D(x) = P{kZ0k ≤ x},x≥0.
Proof. By the definition of the germ-grain model (3.1) we obtain that
PnZ(˜
Fc)∩F6=∅o=Z
N"Y
i:xi∈ψ1−(1 −1˜
F(xi))P{(Z0+xi)∩F6=∅} #P(dψ).
Using the elementary inequality 1 −Q(1 −xi)≤Pxi, 0 ≤xi≤1 , and applying
Campbell’s and Fubini’s theorem lead to
PnZ(˜
Fc)∩F6=∅o≤λZ
˜
Fc
Px∈ˇ
Z0⊕Fdx =λE|˜
Fc∩(F⊕ˇ
Z0)|.
Since ˇ
Z0⊕F⊆[−(a+kZ0k), a +kZ0k]d,we can continue with
PnZ(˜
Fc)∩F6=∅o≤λE−(a+ ∆), a + ∆dc\−(a+kZ0k), a +kZ0k
=λ2d
∞
Z
∆
((a+x)d−(a+ ∆)d)dD(x)
14

≤λd2da+ ∆
∆d−1∞
Z
∆
xddD(x).
Similarly,
PnZ(˜
Gc)∩G6=∅o≤λE−(b−∆), b −∆d\−(b− kZ0k), b − kZ0kc
≤λd2db−∆
∆d−1∞
Z
∆
xddD(x).
Thus, by b−∆ = a+ 3∆ , the proof of Lemma 5.2 is completed.
6 Limit Theorems for Associated Random Measures
The ergodic theorem of Nguyen and Zessin [23] can be used to establish the spatial strong
law of large numbers for the measure η(Γ, W ) if W↑Rd, i.e., Wexpands infinitely in all
directions in a regular way, see [4, p. 332].
Theorem 6.1. In addition to the basic assumptions assume that Ψis ergodic (under
d-dimensional shifts, see [4, p. 341]). Then, for any Γ∈U,
|W|−1η(Γ, W )→λ(Γ) a.s. as W↑Rd(6.1)
with λ(Γ) defined in Lemma 4.1.
Proof. First, note that for any K∈ K we have
Z
Rd
HK(Γ, W +x)dx =|W|HK(Γ,Rd).
In [11] it was proved that the ergodicity of Ψ entails the ergodicity of Zprovided (3.2)
is valid. This, in turn, implies that the spatial stochastic process η(Γ, F ) , F∈Bd
0, (Γ is
fixed) is ergodic under d-dimensional shifts. In order to apply Corollary 4.20 in [23], it is
necessary to bound the family η(Γ, F ) , F⊆[0,1)d,F∈Bd
0, by some integrable random
variable Ybeing independent of F. By (3.3) and (3.4),
η(Γ, F )≤X
i:i≥1
HZi(Γ,[0,1)d−Xi) = Y .
Together with Campbell’s theorem,
EY≤λZ
Rd
EHZ0+x(U,[0,1)d)dx =λH(Γ,Rd)<∞
proving (6.1).
15

This theorem, in particular, yields the almost sure convergence of spatial intensities
for extensions of the intrinsic volumes, see also [11] and [33].
Below we formulate a central limit theorem for the finite-dimensional distributions of
the set-indexed sequence
ˆη(Γ, Wn) = |Wn|1/2η(Γ, Wn)
|Wn|−λ(Γ),Γ∈U,(6.2)
where Wndenotes the cube [−n, n)d. For this, we need a suitable formulation of the
β-mixing condition imposed on the underlying point process Ψ. Assume that
βAΨ([−a, a]d),AΨ(Rd\[−(a+ ∆), a + ∆]d)≤a
min(a, ∆)d−1
βΨ(∆) (6.3)
for any a, ∆≥1 , where βΨ(·) is a non-increasing function on [1,∞) . Condition (6.3)
expresses the degree of weak dependence (in terms of the β-mixing coefficient) between
the behaviour of the point process Ψ inside [−a, a]dand [−(a+ ∆), a + ∆]d. Following
the general concept of spatial mixing it is quite natural and even necessary that the
mixing concept depends on both the distance and the size (volume or surface) of the two
separated support sets, where at least one of them must be bounded, see also [5].
Theorem 6.2. In addition to the above basic assumptions suppose that there exists δ > 0
such that
E"X
i:i≥1
HXi+Zi(U,[0,1)d)#2+δ
<∞,(6.4)
EkZ0k2d(1+δ)/δ +ε<∞for some ε > 0,(6.5)
∞
X
n=1
nd−1(βΨ(n))δ/(2+δ)<∞.(6.6)
(If Pi:i≥1HXi+Zi(U,[0,1)d)≤cwith probability 1 for some constant c < ∞, then put
δ=∞and ε= 0 in (6.5) and (6.6).)
Then, for any k-tuple Γ1, . . . , Γk∈U, the random vector (ˆη(Γ1, Wn),...,ˆη(Γk, Wn)) con-
verges in distribution as n→ ∞ to a k-dimensional centred Gaussian random vector
16

(ξ1,...,ξk)with covariances Eξiξj=σ2(Γi,Γj),1≤i≤j≤k, given by
σ2(Γi,Γj) = λZ
(Rd)2
G!
0[τ{y1,y2}]H(Γi, dy1; Γj, dy2)
+λZ
RdZ
(Rd)2
G!
0,x[τ{y1,x+y2}]Φ(x; Γi, dy1; Γj, dy2)γ(2)
red(dx)
+λZ
RdZ
(Rd)2G!
0,x+y1−y2[τ{y1,x+y1}]−G!
0[τy1]G!
0[τy2]Φ(x, Γi, dy1)Φ(−x, Γj, dy2)dx
−λ2Z
RdZ
(Rd)2
G!
0[τy1]G!
0[τy2]˜
Φ(x, Γi, dy1)˜
Φ(−x, Γj, dy2)dx ,
(6.7)
where the function τand the measures H(Γi, dy1; Γj, dy2)and Φ(x; Γi, dy1; Γj, dy2)are
defined in (4.1),(4.8) and (4.9) respectively, and
Φ(x, Γ, W ) = EhHZ0(Γ,(Zc
0+x)∩W))i,(6.8)
˜
Φ(x, Γ, W ) = H(Γ, W )−Φ(x, Γ, W ).(6.9)
In (6.7) all integrals converge absolutely as a consequence of the mixing and moment
conditions (6.4)–(6.6).
Remark 6.1.Conditions (6.4)–(6.6) ensure the convergence of covariances, i.e.,
Ehˆη(Γi, Wn)ˆη(Γj, Wn)i→σ2(Γi,Γj) as n→ ∞ .
Remark 6.2.Let ldenote the smallest integer greater than or equal to 2 + δ. If the total
variation of the kth-order reduced factorial moment measure [4, p. 357] γ(k)
red if finite for
k= 2,...,l, and EHl
Z0(U,Rd)<∞, then (6.4) holds. This is immediately seen from
Lemma 4.2.
Remark 6.3.The β-mixing condition (6.6) can be verified for quite a few classes of point
processes under mild additional assumptions. For example, in the special case of a Poisson
cluster process Ψ (which is a Boolean model with a random discrete a.s. finite typical
grain or cluster Zc) we have by Lemmata 5.1 and 5.2 that
βΨ(t)≤4λd6d−1Ehρd
c1{ρc≥t/2}i
with ρc= sup{kxk:x∈Zc}.
Similar estimates of the β-mixing rate are known for
•dependently thinned (Poisson) point processes (e.g., soft- and hard-core processes)
as defined by Mat´ern and their generalisations, see [27];
17

•Gibbs point processes satisfying Dobrushin’s uniqueness conditions;
•point processes generated by a (Poisson-) Voronoi tessellation of Rd(e.g., vertices,
midpoint of edges), see [13].
Proof of Theorem 6.2. According to the Cram´er-Wold device we need to prove that, for
any (a1,...,ak)∈Rk\ {0}, the sum
Sn=
k
X
j=1
ajˆη(Γj, Wn)
converges weakly as n→ ∞ to a Gaussian random variable ξwith mean zero and variance
σ2=Pk
i,j=1 aiajσ2(Γi,Γj) . In order to apply a central limit theorem for stationary β-
mixing random fields in [13] we rewrite the above sum as
|Wn|1/2Sn=X
z∈In
Xzwith Xz=
k
X
j=1
aj(η(Γj, Ez)−λ(Γj)) ,
where Ez= [0,1)d+zfor z∈In={−n, . . . , 0,...,n−1}d.
From the simple estimate
η(U,[0,1)d)≤X
i:i≥1
HZi+Xi(U,[0,1)d)
together with (6.4) we deduce that E|X0|2+δ<∞. In view of Lemmata 5.1 and 5.2
combined with (6.3) we obtain
βAη(f)([−a, a]d),Aη(f)(Rd\[−(a+ ∆), a + ∆]d)
≤4a+ ∆
2∆ d−1
βΨ(∆/2) + λd2d+1
∞
Z
∆/4
xddD(x)"4a+ ∆
∆d−1
+12a+ ∆
∆d−1#
for all a, ∆≥1 and any bounded measurable function f:U7→ [0,∞) . The right-hand
side of the latter inequality can be bounded by a term of the form
a
min(a, ∆)d−1
βη(∆)
with
βη(∆) = c1βΨ(∆/2) + c2
∞
Z
∆/4
xddD(x)
18

and constants c1and c2depending only on d. In order to verify the β-mixing condition
needed in the central limit theorem in [13] we have to ensure that
∞
X
n=1
nd−1(βη(n))δ/(2+δ)<∞and ndβη(n)→0 as n→ ∞ .
In turn, this follows from (6.5), (6.6) and the fact that βη(·) is a non-increasing function.
If |Xz| ≤ c,z∈In, for some constant c, we need the convergence of the series
∞
X
n=1
nd−1βη(n)
and
∞
X
n=1
nd−1βΨ(n/2) +
∞
X
n=1
nd−1
∞
Z
n/4
xddD(x).
This results from (6.6) for δ=∞and EkZ0k2d<∞, so that the proof of Theorem 6.2
is completed.
7 A Central Limit Theorem for Random Measures
Generated by Boolean Models
From now on, let Ψ be a stationary Poisson process with intensity λ > 0 , i.e., the
corresponding germ-grain model Zis a Boolean model [19, 27]. In this case, by Slivnyak’s
theorem, γ(2)
red(·) vanishes identically and
G!
x1,x2[τ{0,v}] = exp{−λE|Z0∪(Z0+v)|} =q(v)(1 −p)2,(7.1)
where p=P{0∈Z}= 1 −exp{−λE|Z0|} is the volume fraction of Z,
q(v) = C(v)−p2
(1 −p)2+ 1 = exp{λE|Z0∩(Z0+v)|} , v ∈Rd,(7.2)
and C(v) = P{0∈Z, v ∈Z}is the covariance function of Z. Using these formulae
together with (6.8) we can simplify the covariances σ2(Γi,Γj) in Theorem 6.2 as follows:
σ2(Γi,Γj) = λ(1 −p)2Z
RdZ
Rd
q(y1−y2)H(Γi, dy1; Γj, dy2) (7.3)
+λ2(1 −p)2Z
Rdq(x)Φ(x, Γi)Φ(−x, Γj)−H(Γi,Rd)H(Γj,Rd)dx ,
where
Φ(x, Γ) = Φ(x, Γ,Rd) = EhHZ0(Γ, Zc
0+x))i.
19

Taking into account the inequalities
Z
Rdq(x)−1dx ≤λE|Z0|2exp{λE|Z0|}
and
Z
RdΦ(x, Γi)Φ(−x, Γj)−H(Γi,Rd)H(Γj,Rd)dx
≤E|Z0|HZ0(Γi,Rd) + E|Z0|HZ0(Γj,Rd),
we conclude that σ2(Γi,Γj) , 1 ≤i, j ≤k, are finite whenever
E|Z0|2<∞and EH2
Z0(Γi,Rd)<∞for i= 1,...,k. (7.4)
The following Theorem 7.1 restates the main result of the preceding section in case
of a stationary Poisson process of germs under considerably relaxed conditions. In fact,
these conditions are optimal because they are necessary to ensure the existence of the
covariance matrix. This improvement results from a suitable (although somewhat labo-
rious) approximation technique by m-dependent fields which is quite different from that
used in [9, 10] and [12].
Theorem 7.1. If Ψis a stationary Poisson process with intensity λand (7.4) is satisfied,
then (ˆη(Γ1, Wn),...,ˆη(Γk, Wn)) converges in distribution as n→ ∞ to a Gaussian centred
random vector (ξ1,...,ξk)with the covariances Eξiξj=σ2(Γi,Γj),1≤i, j ≤k, given by
(7.3).
Proof. We only need to consider the univariate case k= 1 for some fixed Γ = Γ1∈U.
With the notation introduced in the proof of Theorem 6.2 we put
Sn=|Wn|−1/2X
z∈InhXz−λ(1 −p)H(Γ,Rd)iwith Xz=η(Γ, Ez).
Setting Fz=Ez⊕Wmfor some fixed integer m≥1 and z∈Zd={0,±1,±2,...}dwe
decompose Sn=S(m)
n+S(m)
n,1+S(m)
n,2by splitting Xzinto three random variables:
X(m)
z=X
i:i≥1
1Fz(Xi)HZi+Xi(Γ, Ez\Ξi(Fz)) with Ξi(B) = [
j:j6=i,Xj∈B
(Zj+Xj),
X(m)
z,1=X
i:i≥1
1Fc
z(Xi)HZi+Xi(Γ, Ez\Ξi(Fz)) ,
X(m)
z,2=X
i:i≥1HZi+Xi(Γ, Ez\Ξi)−HZi+Xi(Γ, Ez\Ξi(Fz)).
20

We first note that, by our assumptions, the independently marked Poisson counting
measures
Ψz=X
i:i≥1
1Ez(Xi)δ[Xi,Zi], z ∈Zd,
can be considered as a family of independent identically distributed random elements tak-
ing values in some measurable space [Nmark,Nmark] of marked counting measures defined
on [0,1)d× K , see [6] for details. Therefore, having in mind the properties of HK(·) ,
K∈ K , it is easily seen that the random variables X(m)
z,z∈In, constitute a stationary
2m-dependent random field which allows a block-representation
X(m)
z=g(Ψy;y∈ {−m,...,m}d+z), z ∈Zd,
where g:N(2m+1)d
mark 7→ R1is a N(2m+1)d
mark -measurable function. Applying the central limit
theorem for this type of weakly dependent fields, see, e.g., [9], yields the weak convergence
of S(m)
n(as n→ ∞) to a centred Gaussian random variable with variance
σ2
m=X
z∈{−m,...,m}d
Cov(X(m)
0, X(m)
z)
provided that E(X(m)
0)2<∞. The latter holds for any m≥1 , if E|Z0|2<∞and
EH2
Z0(Γ,Rd)<∞. In order to prove the asymptotic normality of Snit remains to verify
that
sup
n≥1
E(S(m)
n,i )2≤s(m)
i=X
z∈ZdCov(X(m)
0,i X(m)
z,i )−→ 0 as m→ ∞ for i= 1,2.
After straightforward calculations similar to those leading to (7.3) followed by some
obvious estimates we obtain
|Cov(X(m)
0,1, X(m)
z,1)| ≤ λZ
Fc
0∩Fc
z
EhHZ0(Γ, E0−x)HZ0(Γ, Ez−x)idx
+λ2Z
Fc
0Z
Fc
zZ
KZ
KZ
RdZ
Rd
1E0(x1+y1)1Ez(x2+y2)a(Rd, K2) + b(Rd, K1)+
+ 1 −exp{−w(F0, Fz)}HK1(Γ, dy1)HK2(Γ, dy2)Q(dK1)Q(dK2)dx2dx1,
where
a(F0, K2) = 1F0∩(ˇ
K2+x1+y1)(x2), b(Fz, K1) = 1Fz∩(ˇ
K1+x2+y2)(x1),
and
w(F0, Fz) = λE|F0∩(ˇ
Z0+x1+y1)∩Fz∩(ˇ
Z0+x2+y2)|.
21

Therefore,
s(m)
1≤λZ
Fc
0
EhHZ0(Γ, E0−x)HZ0(Γ,Rd)idx + 2λ2Z
Fc
0
Eh|Z0|HZ0(Γ, E0−x)idx
+λ3E|Z0|2H(Γ,Rd)Z
Fc
0
H(Γ, E0−x)dx .
In view of our moment assumptions (7.4), it follows from the Lebesgue dominated con-
vergence theorem that the right-hand side tends to zero as m→ ∞ .
Next we estimate the covariances occurring in s(m)
2. For notational simplicity write
u(F0) = λE|(F0−(x1+y1)) ∩ˇ
Z0|, v(Fz) = λE|(Fz−(x2+y2)) ∩ˇ
Z0|,
and
I(F0, Fz) = exp{−(u(F0)+v(Fz))}h(1−a(F0, K2))(1−b(Fz, K1)) exp{w(F0, Fz)} − 1i.
Then
|Cov(X(m)
0,2, X(m)
z,2)| ≤ λEZ
RdZ
RdZ
Rd
1E0(x+y1)1Ez(x+y2)
×exp{−λE|(F0−(x+y1)) ∩ˇ
Z0|} − exp{−λE|ˇ
Z0|}HZ0(Γ, dy1)HZ0(Γ, dy2)dx
+λ2Z
RdZ
RdZ
KZ
KZ
RdZ
Rd
1E0(x1+y1)1Ez(x2+y2)I(Rd,Rd)−I(Rd, Fz)
−I(F0,Rd) + I(F0, Fz)HK1(Γ, dy1)HK2(Γ, dy2)Q(dK1)Q(dK2)dx2dx1
≤λ2E|(Fc
0⊕ˇ
E0)∩ˇ
Z0|Z
Rd
EHZ0(Γ, E0−x)HZ0(Γ, Ez−x)dx
+λ2Z
RdZ
RdZ
KZ
KZ
RdZ
Rd
1E0(x1+y1)1Ez(x2+y2)u(Fc
0)v(Fc
z) [w(F0, Fz) + a(F0, K2)
+b(Fz, K1)] + w(Fc
0, F c
z) + [a(Fc
0, K2) + w(Fc
0, Fz)] [b(Fc
z, K1) + v(Fc
z) + w(F0, F c
z)]
+u(Fc
0) [b(Fc
z, K1) + w(F0, F c
z)]HK1(Γ, dy1)HK2(Γ, dy2)Q(dK1)Q(dK2)dx2dx1.
Therefore,
s(m)
2≤λ2E|(Fc
0⊕ˇ
E0)∩ˇ
Z0|EH2
Z0(Γ,Rd)
+ (λE|(Fc
0⊕ˇ
E0)∩ˇ
Z0|)2(λH(Γ,Rd)2E|Z0|2+ 2 H(Γ,Rd)E|Z0|HZ0(Γ,Rd)
+E|(Fc
0⊕ˇ
E0)∩ˇ
Z0|HZ0(Γ,Rd) (1 + 2λE|Z0|)H(Γ,Rd)
+E|(Fc
0⊕ˇ
E0)∩ˇ
Z0||Z0|(2λ+ 3λ2E|Z0|)(H(Γ,Rd))2,
22

whence, arguing as above, we get s(m)
2→0 as m→ ∞.
Corollary 7.2. If Ψis a stationary Poisson process with intensity λ, and Hadmits
integral representation (3.9) such that E|Z0|2<∞and EΥ2
Z0(Γi)<∞for i= 1,...,k ,
then (ˆη(Γ1, Wn),...,ˆη(Γk, Wn)) converges in distribution as n→ ∞ to a Gaussian centred
random vector (ξ1,...,ξk)with
Eξiξj=σ2(Γi,Γj) = λ(1 −p)2E"Z
ΓiZ
Γj
q(ζu1,u2)ΥZ0(du1)ΥZ0(du2)#
+λ2(1 −p)2Z
Rdq(x)Φ(x, Γi)Φ(−x, Γj)−EΥZ0(Γi)EΥZ0(Γj)dx ,
where ζu1,u2=ℓ(u1, Z0)−ℓ(u2, Z0)and
Φ(x, Γ) = E
Z
Γ1−1ˇ
Z0+ℓ(u,Z0)(x)ΥZ0(du)
.
In particular, if U={u1,...,uk}consists of kdistinct points, Γi={ui},1≤i≤k ,
and ΥZ0=δu1+···+δukis a deterministic counting measure, then
σ2(Γi,Γj) = λ(1 −p)2E[q(ζu1,u2)] + λ2(1 −p)2Z
Rdq(x)ϕui(x)ϕuj(−x)−1dx
for 1 ≤i, j ≤k, where ϕu(x) = Px /∈ˇ
Z0+ℓ(u, Z0).
8 Examples and Statistical Applications
In this section we consider only Boolean models. Then the Poisson germ process is de-
termined by only one parameter (the intensity λ), while for the typical grain the mean
values of geometric functionals are usually estimated. The law of large numbers for ran-
dom measures associated with the Boolean model is widely used to estimate the model’s
parameters [21, 26, 31]. The corresponding moment methods equations relate the inten-
sities of random measures and the parameters of interest. In many cases the involved
random measures admit the representation (3.7).
Consider a measure HK(·) on [Σ,B(Σ)] which satisfies our basic assumptions. This
measure is used to define the random measure ηassociated with the underlying Boolean
model. Then Lemma 4.1 and Theorem 6.1 yield
η(Γ, W )
|W|(1 −ˆpW)→λH(Γ,Rd) as W↑Rd,(8.1)
23

where ˆpW=|W∩Z|/|W|estimates the volume fraction of Z. To estimate λ, we set Γ = U
and take a measure HK(·) with a known expected total mass H(U,Rd) = EHZ0(U,Rd)
(for example, any probability measure on Σ will do). Then the obtained estimate of λ
can be plugged into (8.1) to estimate H(Γ,Rd) for another HK(·) .
Below we will prove a central limit theorem for the random measure
˜η(Γ, Wn) = |Wn|1/2η(Γ, Wn)
|Wn|(1 −ˆpWn)−λH(Γ,Rd),(8.2)
where Wn= [−n, n)dand n→ ∞ . The following limit theorem allows us to investigate
asymptotic properties of the estimators obtained by the method of moments.
Theorem 8.1. Under conditions of Theorem 7.1, the random vector ˜η(Γ1, Wn),...,
˜η(Γk, Wn)converges in distribution to a centred Gaussian random vector with the covari-
ances
σ2(Γi,Γj) = λZ
(Rd)2
q(y1−y2)H(Γi, dy1; Γj, dy2) + λ2Z
Rd
˜
Φ(x, Γi)˜
Φ(−x, Γj)q(x)dx ,
where i, j = 1,...,k, and ˜
Φ(x, Γ) = ˜
Φ(x, Γ,Rd) = EHZ0(Γ, Z0+x)),see (6.9).
Proof. Similarly to [22, Theorem 5.5], it suffices to calculate the mixed moment of η(Γ, Wn)
and ˆpn= ˆpWn. First, note that
1−ˆpn=|Wn|−1Z
Wn
[1 −1Z(x)] dx =|Wn|−1Z
WnY
j:j≥1
(1 −1Xj+Zj(x))dx .
Therefore,
E[η(Γ, Wn)(1 −ˆpn)] = |Wn|−1Z
Wn
E"X
i:i≥1Z
Rd
1Wn(Xi+y)(1 −1Xi+Zi(x))
×Y
j:j6=i(1 −1Xj+Zj(x))(1 −1Xj+Zj(Xi+y))HZi(Γ, dy)#dx .
The identity
(1 −1Xj+Zj(x))(1 −1Xj+Zj(Xi+y)) = 1ˇ
Zc
j(Xj−Xi+Xi−x))1ˇ
Zc
j(Xj−Xi−y))
24

together with the refined Campbell theorem yield
E[η(Γ, Wn)(1 −ˆpn)]
=λ|Wn|−1Z
WnZ
RdZ
KZ
Rd
1Wn(z+y)(1 −1ˇ
K(z−x))G!
0[τ{x−z,y}]HK(Γ, dy)Q(dK)dzdx .
=λ|Wn|−1Z
WnZ
RdZ
KZ
Rd
1Wn(v+x)1ˇ
Kc(v−y)G!
0[τ{y−v,y}]HK(Γ, du)Q(dK)dvdx
=λZ
Rd
γWn(v)|Wn|−1Z
KZ
Rd
1ˇ
Kc(v−y)G!
0[τ{y−v,y}]HK(Γ, dy)Q(dK)dv .
The calculations above do not refer to the Poisson assumption. If Ψ is Poisson, then (7.1)
yields
E[η(Γ, Wn)(1 −ˆpn)] = λ(1 −p)2Z
Rd
γWn(v)|Wn|−1Z
Rd
(1 −1ˇ
Z0+y(v))H(Γ, dy)q(v)dv .
The proof can be easily accomplished by elementary calculations.
Remark 8.1.It should be noted that our conditions do not yield the convergence of the
moments in (8.1). In fact, E˜η(Γ, Wn) can be infinite.
Corollary 8.2. Assume that Hadmits integral representation (3.9). Then, under condi-
tions of Theorem 7.1, the statement of Theorem 8.1 holds with the covariances
σ2(Γi,Γj) = λE

Z
ΓiZ
Γj
q(ζu1,u2)ΥZ0(du1)ΥZ0(du2)

+λ2Z
Rd
˜
Φ(x, Γi)˜
Φ(−x, Γj)q(x)dx ,
where ζu1,u2=ℓ(u1, Z0)−ℓ(u2, Z0),and
˜
Φ(x, Γ) = E
Z
Γ
1Z0+ℓ(u,Z0)(x)ΥZ0(du)
.
In particular, if U={u1, . . . , un}, and ΥZ0(·)is the deterministic counting measure, then
σ2
uiuj=λEq(ζu1,u2) + λ2Z
Rd
˜ϕui(x) ˜ϕuj(−x)q(x)dx , (8.3)
where ˜ϕu(x) = 1 −ϕu(x) = Px∈ˇ
Z0+ℓ(u, Z0).
Now we consider several examples of measures HK(·) and discuss possible statistical
applications of the above asymptotic theory.
25

Example 8.1.Let HK(Γ, W ) = |K∩W|be independent of Γ , so that Uconsists of a
single point. Then η(Γ, W ) is equal to the Lebesgue measure of the set of points covered
by Zi+Xifor exactly one i. Clearly, ηcannot be computed if only the union-set Zis
observable. If EHZ0(Γ,Rd) = E|Z0|<∞, then Theorem 6.1 yields
|W|−1η(Γ, W )→λ(1 −p)E|Z0|a.s. as W↑Rd.
Furthermore, if E|Z0|2<∞, then ˆη(Γ, Wn) given by (6.2) satisfies the central limit
theorem (Theorem 7.1) with the limiting variance given by
σ2=λ(1 −p)2E
Z
Z0Z
Z0
q(y1−y2)dy1dy2

+λ2(1 −p)2Z
Rdhq(x)EZ0∩(Zc
0+x)EZ0∩(Zc
0−x)−(E|Z0|)2idx .
Note that ηis defined by formula (3.7), which refers to the individual grains from the
underlying germ-grain model. However, only observations of the union-set Zare available
for the statistical analysis. Most of the interesting examples appear in the case when Uis
the unit sphere Sd−1in Rd, and Hadmits the integral representation (3.9). The typical
grain Z0is supposed to be almost surely convex.
Example 8.2.Let HK(·) admit integral representation (3.9) with ΥZ0concentrated at a
single point u∈Uso that ΥZ0({u}) = 1 . Then
HK(Γ, W ) = 1W(ℓ(u, K))1Γ(u)
is a probability measure on Σ. Set Γ = U. Then, the ith term in (3.7) is 1 if and only if
ℓ(u, Zi+Xi)∈W\Ξi. The latter means that the specific point, ℓ(u, Zi+Xi) , of the ith
grain is exposed, i.e., this specific point is not covered by the grains (Zj+Xj) with j6=i.
Then, for all Γ ∋u,η(Γ, W ) is equal to the number of exposed specific points inside W,
i.e.,
η(Γ, W ) = #{i:ℓ(u, Zi) + xi∈W\Ξi}.
For example, if U=Sd−1and ℓ(u, K) is the tangent point of Kin the direction u, then
η(Γ, W ) is the number N+(u, W ) of exposed tangent points in the direction u, see [22, 20].
Note that ℓ(u, K) is the lexicographical minimum of the support set L(u, K ) defined as
L(u, K) = {x∈∂K :hu, xi=−h(K, u)},(8.4)
where hu, xiis the scalar product and h(K, u) = sup{hu, xi:x∈K}is the support
function of K.
26

Then Lemma 4.2 gives moment measures for the point process of tangent points. Since
η(Γ, W ) = N+(u, W ) is observable, and
ˆ
λW=η(Γ, W )
|W|(1 −ˆpW)→λa.s. as W↑Rd,(8.5)
it is possible to estimate the intensity of the Boolean model using the spatial intensity of
η(or the intensity of the point process of exposed tangent points). Corollary 8.2 yields a
central limit theorem for the corresponding intensity estimator. Since ˜ϕu(x) ˜ϕu(−x) = 0
for almost all x, (8.3) yields
σ2
uu =λ
1−p.(8.6)
This variance has been computed directly in [22]. If ΥZ0= Υ0is a deterministic prob-
ability measure on U=Sd−1, then η(U, W ) is the weighted number of exposed tangent
points considered in [20], so that Corollary 8.2 yields Theorem 3.1 of [20], which gives the
asymptotic variance of the corresponding estimator.
Example 8.3.Suppose that the support set L(u, Z0) = {ℓ(u, Z0)}is a singleton for all
u∈U=Sd−1and almost all realisations of Z0. Let kZ0(u) be a positive function which
depends on ∂Z0∩Bε(ℓ(u, Z0)) for arbitrarily small ε > 0 . In particular, kZ0(u) can
be the absolute curvature of ∂Z0at the corresponding tangent point or a function of
this curvature. The measure His given by the integral representation (3.9) with ΥZ0(·)
concentrated at {u}with mass kZ0(u). Then
Eη({u}, W ) = λ|W|(1 −p)EkZ0(u),
whence the expected value of kZ0(u) can be estimated if an estimator of λis available.
For instance, if the grain is a random ball of radius ξ, then all moments of ξand also all
expectations Ef(ξ) (if they exist) can be estimated.
If E|Z0|2<∞and Ek2
Z0(u)<∞for all u, then Corollary 8.2 is applicable with the
limiting variance given by
σ2
uu(k) = λE(kZ0(u)2)/(1 −p).
Example 8.4.Let d= 2 , Ube the unit circle, and let HK(Γ,·) with Γ ∋ube concentrated
at the tangent point ℓ(u, K) with mass 1 if the support set L(u, K ) is a singleton. Other-
wise, L(u, K) is a segment and HK(Γ,·) assigns the weights 1/2 to its end-points denoted
by ℓ′(u, K) and ℓ′′ (u, K) . Note that HK(·)does not admit integral representation (3.9).
The corresponding measure ηis still observable and can be used to estimate the
intensity of the Boolean model, since, by Theorem 6.1, η(Γ, W )/|W| → λ(1 −p) almost
surely as W↑Rd. Theorem 8.1 implies that the asymptotic variance of the estimator ˆ
λW
from (8.5) is equal to
σ2
uu =λE[q(0) + q(ℓ′′(u, Z0)−ℓ′(u, Z0))]/2,
27

where qis defined in (7.2). This variance is less than the variance given by (8.6) of the
usual tangent points estimator, since q(v)≤q(0) = 1/(1 −p) for v∈Rd.
Example 8.5.Assume that U=Sd−1, and HZ0(Γ, W ) = Θd−1(Z0,Γ×W) is the (d−1)-
dimensional generalised surface area measure of the typical grain, see Appendix. If Wis
an open set, then, by Theorem A.1, η(Γ, W ) is equal to the surface area measure of the
boundary of the germ-grain model Zmeasured inside W. In particular, η(Sd−1, W ) is
equal to the surface area of (∂Z)∩W. Then the results above give the ergodic theorem
and the central limit theorem for surface measures. In particular, the ergodic theorem
yields an estimator of the mean surface area measure H(Γ,Rd) = EΘd−1(Z0,Γ×Rd) of
Z0.
If the typical grain has no flat pieces on its boundary, then HK(·) admits integral
representation (3.9) with ℓ(u, K) being the tangent point and ΥZ0(Γ) = Θd−1(Z0,Γ×Rd)
being the area measure of order (d−1) , see [25, p. 203]. Then Theorem 7.1 yields the
limit theorem for surface measures proved in [20] for the bounded grains. Note that it
is possible to extend this example for non-convex grains with rectifiable boundaries. If
the window Wis closed, then the situation is more complicated, since the parts of the
boundary of Wcovered by Zcontribute to η, see [20]. It was shown in [20] that the
central limit theorem does not hold in this case.
The following example was inspired by Hall [7, Section 5.6], who considered the planar
Boolean model with a circular typical grain of radius ξand statistic
κ(W) = X
i:i≥1
ϕia(Ri),(8.7)
where ϕiis the angular content of the exposed boundary of the ith grain (disk) within
the window W,Riis the radius of the ith disk, and a: [0,∞)7→ [0,∞) is a function. In
other words, κis the weighted sum of the angular contents of all protruded pieces of the
boundary. It is shown in [7, p. 323] that
Eκ(W) = 2πλ|W|(1 −p)Ea(ξ),(8.8)
if Eξ2<∞and E|a(ξ)|<∞, where ξis the radius of the typical grain (disk). From
this, it is possible to estimate Ea(ξ).
Example 8.6.Let HK(Γ, W ) = Θj(K, Γ×W)f(K) , where f(K) is a translation-invariant
positive functional on the space of convex compact sets and Θjis the jth generalised
curvature measure, see [25] and (A.1). The corresponding measure η(·) is observable as
soon as the value f(K) is retrievable from any relatively open piece of the boundary ∂K .
In particular, this is true if the typical grain is a ball. It follows from Lemma 4.1 that
Eη(Γ, W ) = λ|W|(1 −p)EhΘj(Z0,Γ×Rd)f(Z0)i,
28

which yields (8.8) as a special case for d= 2, Γ = U=Sd−1,j= 1, and f(Bξ(x)) =
ξ−1a(ξ) .
If E|Z0|2<∞and EVj(Z0)f(Z0)2<∞(where Vjis the intrinsic volume, see [25,
p. 210]), then Theorem 7.1 implies that the finite-dimensional distributions of ˜η(Γ, Wn) are
asymptotically Gaussian with the covariance (7.3). These moment assumptions coincide
with those imposed in [7, Theorem 5.3] for planar Boolean models with circular grains.
In the latter case HZ0(·) admits integral representation (3.9) with ΥZ0(du) = a(ξ)du and
ℓ(u, Z0) being the tangent point of Z0in direction u. Then Corollary 7.2 yields the
asymptotic variance of κ(W)
σ2
κ=λ(1 −p)2E
a(ξ)2ξ−2Z
∂Z0Z
∂Z0
q(y1−y2)dy1dy2

+λ2(1 −p)2Z
R2q(x)b(x)b(−x)−4π2(Ea(ξ))2dx ,
where the integrals over curves are understood with respect to the 1-dimensional Hausdorff
measure (curve length), and
b(x) = E

a(ξ)ξ−1Z
∂Z0∩(Zc
0+x)
dy


is the expected angular content of ∂Z0within Zc
0+xmultiplied by a(ξ) .
Example 8.7.Let Z0be a random convex polytope in Rd. Assume that U=Sd−1and
HZ0(U, W ) is the number of vertices of Z0lying inside W. Then η(U, W ) gives the number
of exposed vertices inside W. Theorem 6.1 yields
η(U, W )/|W| → λ(1 −p)Eνas W↑Rd,
where νis the number of vertices of the typical grain Z0. The central limit theorem is
valid if both νand |Z0|have finite second moments. In order to compute the limiting
variance consider the vertices of Z0to be a point process ΨZ0with a finite total number
of points. Its second-order moment measure is denoted by α(2)
Z0. Then (7.3) yields
σ2
UU =λ(1 −p)2Z
(Rd)2
q(x−y)α(2)
Z0(d(x, y))
+λ2(1 −p)2Z
Rd
(q(x)f(x)f(−x)−(Eν)2)dx , (8.9)
where
f(x) = EX
Yi∈ΨZ0
1Zc
0(Yi−x).
29

Example 8.8.Let H(r)
Z0(Γ, W ) = ¯µr(Z0,Γ×W) , see (A.1). Then the corresponding measure
η(r)(Γ, W ) coincides with ¯µr(Z, Γ×W) . As in (A.1),
η(r)(Γ, W ) = 1
d
d−1
X
j=0
rd−jd
jηj(Γ, W ),(8.10)
where ηj(Γ, W ) , j= 0, . . . , d−1 , are measures defined by (3.7) for Hj
K(Γ, W ) = Θj(K, Γ×
W) , K∈ C . We can use (8.2) to define the corresponding measures ˜ηj(Γ, Wn) . Assume
that E|Z0⊕Br(0)|2<∞for some r > 0.
Note that, similarly to the standard Cram´er-Wold device, a random vector
(˜η0(Γ, Wn),...,˜ηd−1(Γ, Wn))
converges in distribution to (ξ0,...,ξd−1) as n→ ∞ , if, for each r > 0 ,
d−1
X
j=0
rj˜ηj(Γ, Wn)
converges to Pd−1
j=0 rjξjweakly as n→ ∞ . After applying this to the polynomial expan-
sions (8.10) and (A.1), one can prove that the random vector (˜η0(Γ, Wn),...,˜ηd−1(Γ, Wn))
converges in distribution to a centred Gaussian random vector with the covariances given
by
σij =λZ
(Rd)2
q(y1−y2)Hij(Γ, dy1; Γ, dy2) + λ2Z
Rd
˜
Φi(x, Γ)˜
Φj(−x, Γ)q(x)dx ,
where i, j = 0,...,d−1 ,
Hij (Γ, W1; Γ, W2) = EhΘi(Z0,Γ×W1)Θj(Z0,Γ×W1)i,
and
˜
Φi(x, Γ) = EΘi(Z0,Γ×(Z0+x)).
This example allows to find the joint limit distribution for the estimators of the intensities
of Minkowski measures of different orders.
9 Concluding Remarks
To conclude with, we outline several possible generalisations. We give only the results for
the first moments in the Poisson case, although a laborious application of the methods
developed above (with evident changes) allows to derive the corresponding limit theorems.
30

Let us consider a measure HK1,K2(·) which depends on two compact sets K1and K2
in such a way that, for all x∈Rd,
HK1+x,K2+x(Γ, W +x) = HK1,K2(Γ, W ) = HK1,K2(Γ, W ∩K1∩K2).(9.1)
Define
η(Γ, W ) = X
i,j≥1, i6=j
HZi+Xi,Zj+Xj(Γ, W \Ξij),
where
Ξij =[
k:k6=i,j
(Xk+Zk).
Then the first moment of ηis given by
Eη(Γ, W ) = λ|W|Z
RdZ
Rd
G!
0,z[τy]Hz(Γ, dy)α(2)
red(dz),
where
Hz(Γ, W ) = EHZ1,Z2+z(Γ, W )
for two independent grains Z1and Z2(we assume that Hz(Γ,Rd) is finite for all z∈Rd).
If the point process of germs is Poisson, then
Eη(Γ, W ) = λ2(1 −p)|W|E
Z
Rd
HZ1,Z2+z(Γ,Rd)dz
.(9.2)
Example 9.1.Let Zbe the Boolean model with a.s. convex grains. For Γ = Uconsider
the measure
HK1,K2(Γ, W ) = 1∂ K1∩∂K2∩W6=∅, K1, K2∈ C ,
which satisfies (9.1). Then,
Hz(Γ,Rd) = Z
Rd
1∂Z1∩(∂Z2+z)6=∅dz =|∂Z1⊕∂ˇ
Z2|,
whence
Eη(Γ, W ) = λ(1 −p)|W|E|∂Z1⊕∂ˇ
Z2|.(9.3)
For instance, if Z0=Mis a deterministic central symmetric convex set, then ∂M ⊕∂ˇ
M=
2Mand (9.3) yields
Eη(Γ, W ) = λ(1 −p)2d|W||M|.
To give a geometric interpretation of the measure ηin the planar case, remember that
Xi+∂Ziand Xj+∂Zjare either disjoint or intersect at two points (they may touch with
probability zero). Then 2η(Sd−1, W ) gives the number of such points in W(resulted from
(Xi+∂Zi)∩(Xj+∂Zj)) which are exposed (not covered by Xk+Zkwith k6=i, j).
31

Example 9.2.Let Zbe the planar Boolean model with almost surely strictly convex typical
grain Z0(so that ∂Z0does not contain a segment for almost all realisations of Z0). Then
L(u, Z0) = {ℓ(u, Z0)}a.s. for all u∈U=Sd−1. For strictly convex K1and K2set
HK1,K2({u}, W ) = 1K1∩K2∩W6=∅1ℓ(−u,K1∩K2)∈W1ℓ(−u,K1)/∈K21ℓ(−u,K2)/∈K1, u ∈U.
Then η({u}, W ) equals the number of negative tangent points in direction u, see [27,
p. 241]. The difference between the number of positive tangent points (Example 8.2) and
η({u}, W ) is equal to the Euler-Poincar´e characteristic of Z∩Int W, see [26, 27].
Now (9.2) yields
Eη({u}, W ) = λ2(1 −p)|W|EZ
Rd
1Z1∩(Z2+z)6=∅1ℓ(−u,Z1)/∈Z2+z1ℓ(−u,Z2)+z /∈Z1dz
=λ2(1 −p)|W|E(Z1⊕ˇ
Z2)\[(ℓ(−u, Z1)⊕ˇ
Z2)∪(Z1⊕ℓ(u, ˇ
Z2))]
=λ2(1 −p)|W|E|Z1⊕ˇ
Z2| − E|Z1| − E|Z2|,
since ℓ(−u, Z1)∈Z1and ℓ(u, ˇ
Z2)∈ˇ
Z2. It is known ([25, p. 275] and [32]) that
|Z1⊕ˇ
Z2|=|Z1|+|Z2|+ 2A(Z1,ˇ
Z2),
where
A(Z1,ˇ
Z2) = 1
2Z
S1
h(Z1, u)Θ1(ˇ
Z2, du ×Rd)
is a so-called mixed area of Z1and ˇ
Z2. Since Z1and Z2are independent and have the
same distribution as Z0,
EA(Z1,ˇ
Z2) = A(EZ0,Eˇ
Z0),
where EZ0is the Aumann expectation of Z0, which satisfies Eh(Z0, u) = h(EZ0, u) and
EΘ1(ˇ
Z0, du ×Rd) = Θ1(Eˇ
Z0, du ×Rd), see [28, 31, 32]. Therefore,
Eη({u}, W ) = λ2(1 −p)|W|A(EZ0,Eˇ
Z0).
From this we immediately obtain the spatial density of the Euler-Poincar´e characteristic:
χ(Z∩W)
|W|→λ(1 −p)(1 −λA(EZ0,Eˇ
Z0)) a.s. as W↑Rd,
which was first derived by Weil [31]. For general convex Z0this result can be obtained by
approximations with strictly convex sets. The technique above can be applied to prove
the central limit theorem for the Euler-Poincar´e characteristics. It will be considered
elsewhere.
32

Clearly, it is possible to consider also vector-valued measures HK(·) . In fact, many
results can be generalised also for the germ-grain model Zgenerated by a not neces-
sarily independently marked point process. Unfortunately, in this case formulae for the
variances are getting incomprehensible. Note that other generalisations are possible for
independently marked point processes with other spaces of marks, for example, when the
points are marked by functions, measures or capacities.
Acknowledgements
The authors are grateful to Centrum voor Wiskunde en Informatica (CWI, Amsterdam)
for hospitality. I. Molchanov was supported in part by the Netherlands Organisation for
Scientific Research (NWO).
Appendix
Random Measures and Geometric Functionals
Curvature measures are very important functionals defined on the family of convex sets,
see [25]. The positive extension of curvature measures onto the convex ring R(the family
of finite unions of convex compact sets) is defined as follows, see [24, 25, 30] for further
details. For given F∈ R and x∈Rd, a point q∈Fis called a projection of xonto Fif
there exists a neighbourhood Gof qsuch that qis the nearest point to xamong all points
from F∩G. Let Π(F, x) be the set of all projections of x. For Γ ⊆Sd−1and r > 0 let
¯cr(F, Γ×W, x) be the number of points q∈Π(F, x)∩Wsuch that 0 <kx−qk ≤ rand
the direction of x−qbelongs to Γ. Then
¯µr(F, Γ×W) = Z
Rd
¯cr(F, Γ×W, x)dx =1
d
d−1
X
j=0
rd−jd
j¯
Θj(F, Γ×W),(A.1)
see [24]. If K∈ C , then we write Θj(K, Γ×W) instead of ¯
Θj(K, Γ×W), which is called
the jth generalised curvature measure of K. For F∈ R , the coefficients ¯
Θj(F, Γ×W)
in (A.1) are called the positive extensions of the curvature measures.
Formula (3.7) suggests another way to define functionals on the convex ring. Let
HK(·) be a measure on [Σ,B(Σ)] , where K∈ C . Then
ηF(Γ, W ) =
n
X
i=1
HKi(Γ, W \ ∪1≤j≤n,j6=iKj).(A.2)
extends Hfor F=∪n
i=1Ki. In general, this extension depends on the decomposition of F
into the union of convex sets. Indeed, for any convex Fwe obtain ηF(Γ, W ) = 0 by using
33

the trivial representation F=F∪F. However, as it will be shown, such a situation is not
possible if Fis a realisation of a germ-grain model satisfying rather weak assumptions.
If F∈ R , then x∈∂F is said to be an exposed positive tangent point if x= Π(F, v)
for some v /∈F. The set of all exposed positive tangent points of F∈ R is denoted
by ∂+Fand is said to be the positive boundary of F. The set-difference ∂F \∂+Fis
comprised of sets of dimensions not greater than (d−2) . Note that ∂+Fcontains the set
D(K1,...,Kn) =
n
[
i=1
(∂Ki\ ∪1≤j≤n,j 6=iKj)
for each decomposition F=∪n
i=1Kiof Finto the union of convex compact sets.
Let us use (A.2) to extend onto Rthe measure HK(Γ, W ) = ¯µr(K, Γ×W) , where
U=Sd−1. For the moment, we assume that F=∪n
i=1Kiwith
D(K1,...,Kn) = ∂+F . (A.3)
This means that the positive boundary of Fis equal to the union of all ‘visible’ (or
exposed) boundaries of individual grains. A similar condition appears in [36] when con-
sidering unions of sets of positive reach. By (A.1) and (A.3),
¯µr(F, Γ×W) =
n
X
i=1
HKi(Γ, W \ ∪1≤j≤n,j 6=iKj).
Expanding both sides in the polynomials (A.1) shows that the basic formula (A.2) applied
to the curvature measure HK(Γ, W ) = Θj(K, Γ×W) , K∈ C , gives its positive extension
onto the convex ring, i.e., ηF(Γ, W ) = ¯
Θj(F, Γ×W) .
In the following we give conditions, when the identity (A.3) holds for the germ-grain
model (3.1). First, define the set-theoretic limit
D(Xi+Zi;i≥1) = lim
n→∞ D(Xi1+Zi1,...,Xik(n)+Zik(n)) = [
i:i≥1
(∂(Xi+Zi)\Ξi),
where {i1,...,ik(n)}={i≥1 : Xi∈Bn(0)}.
Theorem A.1. Let Ψbe a stationary second-order point process with second-order re-
duced factorial moment measure α(2)
red(·), which is absolutely continuous with respect to
the Lebesgue measure in Rd. Assume that the typical grain Z0is a.s. compact and
convex. Then, for Zdefined in (3.1), we have
PD(Xi+Zi;i≥1) = ∂+Z= 1 .
Clearly, the conditions of Theorem A.1 hold for each Boolean model with a convex
typical grain.
34

Proof. Let K1and K2be two convex compact sets. Then D(K1, K2)6=∂+(K1∪K2)
implies that either L(u, K1)∩L(u, K2)6=∅or L(u, K1)∩L(−u, K2)6=∅for some u∈Sd−1.
Thus,
L(K1, K2) = {(x, y) : D(K1+x, K2+y)6=∂+((K1+x)∪(K2+y))}
⊆[
u∈Sd−1n(x, y) : y−x∈(L(u, K1)⊕ˇ
L(u, K2)) ∪(L(u, K1)⊕ˇ
L(−u, K2))o.
It follows from Theorem 1.7.5 of [25] that
L(u, K1)⊕ˇ
L(−u, K2) = L(u, K1)⊕ L(u, ˇ
K2) = L(u, K1⊕ˇ
K2).
Thus,
L(K1, K2)⊆ {(x, y) : y−x∈∂(K1⊕ˇ
K2)∪Λ(K1, K2)},(A.4)
where
Λ(K1, K2) = [
u∈Sd−1
[L(u, K1)⊕ˇ
L(u, K2)] ,
see [25, p. 86]. The technique described in [25, Section 2.3] can be used to prove that the
Lebesgue measure of Λ(K1, K2) is equal to zero. First, the equality L(u, K1⊕B1(0)) =
L(u, K1)+uallows to consider sets K1and K2which contain a ball of radius 1. Therefore,
Lemma 2.3.9 of [25] yields
Λ(K1, K2)⊂
m
[
i=1
((Ci+ˇ
Ci) + ai),(A.5)
where ai∈Rdand C1,...,Cmare caps of K1⊕K2covering the boundary of K1⊕K2. (A
cap is defined to be a non-empty intersection of the convex set with a closed half-space.)
Furthermore, Theorem 2.3.2 [25] gives a possibility to choose these caps in such a way
that m
X
i=1
|Ci|< ε
for any given ε > 0 . Note that |Ci⊕ˇ
Ci| ≤ (d+ 1)d|Ci|. The latter follows from the fact
that ˇ
C⊂dC for any convex compact set Cwith non-empty interior and having its centroid
at the origin [25, p. 81]. Thus, the Lebesgue measure of the set in the right-hand side of
(A.5) can be made arbitrarily small. By (A.4), L(K1, K2)⊆ {(x, y) : y−x∈˜
L(K1, K2)}
with |˜
L(K1, K2)|= 0 .
Now consider the germ-grain model Zdefined by (3.1). Notice that
{D(Xi+Zi;i≥1) 6=∂+Z}
⊆[
i6=j
{D(Zi+Xi, Zj+Xj)6=∂+((Zi+Xi)∪(Zj+Xj))}
=[
i6=j
{(Xi−Xj)∈˜
L(Zi, Zj)}.
35

The probability of the event in the right-hand side equals the limit (as n→ ∞) of the
probabilities
P

[
i6=j:Xi,Xj∈Bn(0)
{Xi−Xj∈˜
L(Zi, Zj)}


≤Z
Bn(0)×Bn(0) Z
K×K
1˜
L(K1,K2)(x−y)Q(dK1)Q(dK2)α(2)(d(x, y))
=λZ
C×C Z
Bn(0)
α(2)
red(˜
L(K1, K2)∩Bn(y))dyQ(dK1)Q(dK2).
By the assumptions of Theorem A.1, α(2)
red(˜
L(K1, K2)) = 0, so that the latter integral is
equal to zero for every n≥1. This completes the proof of Theorem A.1.
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