Abstract
First we put together basic definitions and fundamental facts and results from the theory of (un)marked point processes defined on Euclidean spaces \({\mathbb{R}}^{d}\). We introduce the notion random marked point process together with the concept of Palm distributions in a rigorous way followed by the definitions of factorial moment and cumulant measures and characteristics related with them. In the second part we define a variety of estimators of second-order characteristics and other so-called summary statistics of stationary point processes based on observations on a “convex averaging sequence” of windows \(\{W_{n},\,n \in \mathbb{N}\}\). Although all these (mostly edge-corrected) estimators make sense for fixed bounded windows our main issue is to study their behaviour when W n grows unboundedly as n → ∞. The first problem of large-domain statistics is to find conditions ensuring strong or at least mean-square consistency as n → ∞ under ergodicity or other mild mixing conditions put on the underlying point process. The third part contains weak convergence results obtained by exhausting strong mixing conditions or even m-dependence of spatial random fields generated by Poisson-based point processes. To illustrate the usefulness of asymptotic methods we give two Kolmogorov–Smirnov-type tests based on K-functions to check complete spatial randomness of a given point pattern in \({\mathbb{R}}^{d}\).
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4.1 Marked Point Processes: An Introduction
First we present a rigorous definition of the marked point process on Euclidean spaces with marks in some Polish space and formulate an existence theorem for marked point processes based on their finite-dimensional distributions. Further, all essential notions and tools of point process theory such as factorial moment and cumulant measures with their densities and reduced versions as well as the machinery of Palm distributions in the marked and unmarked case are considered in detail.
4.1.1 Marked Point Processes: Definitions and Basic Facts
Point processes are mathematical models for irregular point patterns formed by randomly scattered points in some locally compact Hausdorff space. Throughout this chapter, this space will be the Euclidean space \({\mathbb{R}}^{d}\) of dimension \(d \in \mathbb{N}\). In many applications to each point X i of the pattern a further random element M i , called mark, can be assigned which carries additional information and may take values in a rather general mark space M equipped with an appropriate σ-algebra \(\mathcal{M}\). For example, for d = 1, the X i ’s could be arrival times of customers and the M i ’s their sojourn times in a queueing system and, for d = 2, one can interpret the X i ’s as locations of trees in a forest with the associated random vectors M i of stem diameter, stem height and distance to the nearest-neighbour tree. In this way we are led to the notion of a (random) marked point process which can be adequately modeled as random counting measure Ψ M ( ⋅) on Cartesian products B ×L, which gives the total number of points in a bounded subset B of \({\mathbb{R}}^{d}\) whose marks belong to a set of marks L ⊆ M. To be precise, we need some further notation. Let \(\mathcal{N}_{\mathsf{M}}\) denote the set of locally finite counting measures ψ( ⋅) on the measurable product space \(({\mathbb{R}}^{d} \times \mathsf{M},\,\mathcal{B}({\mathbb{R}}^{d}) \otimes \mathcal{M})\), i.e. \(\psi \in \mathcal{N}_{\mathsf{M}}\) is a σ-additive set function on \(\mathcal{B}({\mathbb{R}}^{d}) \otimes \mathcal{M}\) taking on non-negative integer values such that ψ(B ×M) < ∞ for any bounded Borel set \(B \in \mathcal{B}({\mathbb{R}}^{d})\). We then define \(\mathfrak{N}_{\mathsf{M}}\) to be the smallest σ-algebra containing all sets \(\{\psi \in \mathcal{N}_{\mathsf{M}} : \psi (B \times L) = n\}\) for \(n \in \mathbb{N} \cup \{ 0\}\), any bounded \(B \in \mathcal{B}({\mathbb{R}}^{d})\) and \(L \in \mathcal{M}\). Finally, let \((\Omega ,\mathcal{F},\mathbf{P})\) be a hypothetical probability space on which all subsequent random elements will be defined.
Definition 4.1.
A \((\mathcal{F},\mathfrak{N}_{\mathsf{M}})\)-measurable mapping
is said to be a (random) marked point process (briefly: MPP) on \({\mathbb{R}}^{d}\) with mark space \((\mathsf{M},\mathcal{M})\). In other words, a MPP Ψ M ( ⋅) (ω will be mostly suppressed) is a random locally finite counting measure on \(({\mathbb{R}}^{d} \times \mathsf{M},\mathcal{B}({\mathbb{R}}^{d}) \otimes \mathcal{M})\).
The probability measure \(P_{\mathsf{M}}(A) = \mathbf{P}(\{\omega \in \Omega : \Psi _{\mathsf{M}}(\omega ,\cdot ) \in A\})\) for \(A \in \mathfrak{N}_{\mathsf{M}}\) induced on \((\mathcal{N}_{\mathsf{M}},\mathfrak{N}_{\mathsf{M}})\) is called the distribution of Ψ M —briefly expressed by \(\Psi _{\mathsf{M}} \sim P_{\mathsf{M}}\). Here and in what follows we put Ψ( ⋅) = Ψ M ( ⋅ ×M) to denote the corresponding unmarked point process and write in general Ψ ∼ P to indicate point processes without marks. One often uses the notation
where δ x (A) = 1 for x ∈ A and δ x (A) = 0 for x∉A (Dirac measure). Note that due to the local finiteness of Ψ M there are at most countably many atoms but each atom occurs with random (P-a.s. finite) multiplicity. The indexing in (4.1) does not need to be unique and the X i ’s occur in the sums according to their multiplicity. In accordance with the general theory of random processes our next result formulates an analogue to Kolmogorov’s extension theorem stating the existence of a probability space \((\Omega ,\mathcal{F},\mathbf{P})\) in Definition 4.1 in case of Polish mark spaces M given the family of finite-dimensional distributions \(\mathbf{P}(\Psi _{\mathsf{M}}(B_{1} \times L_{1}) = n_{1},\ldots ,\Psi _{\mathsf{M}}(B_{k} \times L_{k}) = n_{k})\).
Theorem 4.1.
Let M be a Polish space equipped with the corresponding Borel-σ-algebra \(\mathcal{B}(\mathsf{M})\) generated by a family \(\mathcal{M}_{0}\) of subsets in M such that \(\mathcal{S} = \big\{\times _{i=1}^{d}[a_{i},b_{i}) \times L : -\infty < a_{i} \leq b_{i} < \infty \,,\,L \in \mathcal{M}_{0}\big\}\) is a semi-ring which generates the ring of bounded Borel sets in \({\mathbb{R}}^{d} \times \mathsf{M}\) and each bounded Borel set \(X \subset {\mathbb{R}}^{d} \times \mathsf{M}\) can be covered by a finite sequence \(S_{1},\ldots ,S_{m} \in \mathcal{S}\) .
For any finite sequence of pairwise disjoint \(B_{1} \times L_{1},\ldots ,B_{k} \times L_{k} \in \mathcal{S}\) define the distribution
of a k-dimensional random vector with non-negative integer-valued components.
Then there exists a unique probability measure P M on the measurable space \((\mathcal{N}_{\mathsf{M}},\mathfrak{N}_{\mathsf{M}})\) with finite-dimensional distributions
for \(n_{1},\ldots ,n_{k} \in \mathbb{N} \cup \{ 0\}\) and any \(k \in \mathbb{N}\,\) , if the following conditions for the family of probabilities \(p_{n_{1},\ldots ,n_{k}}(B_{1} \times L_{1},\ldots ,B_{k} \times L_{k})\) are satisfied:
-
1.
Symmetry:
$$\begin{array}{rcl} p_{n_{1},\ldots ,n_{k}}(B_{1} \times L_{1},\ldots ,B_{k} \times L_{k}) = p_{n_{\pi (1)},\ldots ,n_{\pi (k)}}(B_{\pi (1)} \times L_{\pi (1)},\ldots ,B_{\pi (k)} \times L_{\pi (k)})& & \\ \end{array}$$for any permutation π :{ 1,…,k}↦{1,…,k} ,
-
2.
Consistency:
$$\begin{array}{rcl} & & \sum \limits _{n=0}^{\infty }p_{ n_{1},\ldots ,n_{k-1},n}(B_{1} \times L_{1},\ldots ,B_{k-1} \times L_{k-1},B_{k} \times L_{k}) \\ & & \quad = p_{n_{1},\ldots ,n_{k-1}}(B_{1} \times L_{1},\ldots ,B_{k-1} \times L_{k-1})\,, \\ \end{array}$$ -
3.
Additivity: If \(B_{j} \times L_{j} \cup \cdots \cup B_{k} \times L_{k} \in \mathcal{S}\,\) , then
$$\begin{array}{rcl} & & \sum \limits _{\begin{array}{c}n_{j}+\cdots +n_{k}=n \\ n_{j},\ldots ,n_{k}\geq 0 \end{array}}p_{n_{1},\ldots n_{j-1},n_{j},\ldots ,n_{k}}(B_{1} \times L_{1},\ldots ,B_{j-1} \times L_{j-1},B_{j} \times L_{j},\ldots ,B_{k} \times L_{k}) \\ & & \quad = p_{n_{1},\ldots ,n_{j-1},n}(B_{1} \times L_{1},\ldots ,B_{j-1} \times L_{j-1},(B_{j} \times L_{j} \cup \cdots \cup B_{k} \times L_{k})) \\ \end{array}$$for j = 1,…,k.
-
4.
Continuity: For any sequence of pairwise disjoint sets \(B_{j}^{(n)} \times L_{j}^{(n)} \in \mathcal{S}\) for j = 1,…,k n with k n ↑∞ satisfying \(\bigcup \nolimits _{j=1}^{k_{n}}(B_{j}^{(n)} \times L_{j}^{(n)}) \downarrow \varnothing \,\) , it holds \(p_{0,\ldots ,0}(B_{1}^{(n)} \times L_{1}^{(n)},\ldots ,B_{k_{n}}^{(n)} \times L_{k_{n}}^{(n)}) \uparrow 1\) as n →∞ .If M ={ m} consists of a single (or at most finitely many) element(s), then the latter condition can be replaced by
$$\begin{array}{rcl} & & \lim \limits _{n\rightarrow \infty }\;p_{0}(\times _{i=1}^{j-1}[a_{ i},b_{i}) \times [x_{j} - \frac{1} {n},x_{j}) \times \times _{i=j+1}^{d}[a_{ i},b_{i}) \times \{ m\}) = 1 \\ & & \mbox{ for all}\:\:{(x_{1},\ldots ,x_{d})}^{\top }\in {\mathbb{R}}^{d}\:\:\mbox{ and}\:\: -\infty < a_{ i} < b_{i} < \infty \:,\:i = 1,\ldots ,d\,\end{array}$$
This allows a canonical choice of the probability space \((\Omega ,\mathcal{F},\mathbf{P})\) mapping Ψ M :
Remark 4.1.
There exists a metric τ in the set \(\mathcal{N}_{\mathsf{M}}\) such that the metric space \((\mathcal{N}_{\mathsf{M}},\tau )\) is separable and complete whose Borel-σ-algebra \(\mathcal{B}(\mathcal{N}_{\mathsf{M}})\) coincides with \(\mathfrak{N}_{\mathsf{M}}\,\). This allows to introduce the notion of weak convergence of MPP’s in the usual way, see [69].
To prove Theorem 4.1 one has only to reformulate the proof of a corresponding result for (unmarked) PP’s on Polish spaces in [346]. Readers interested in more background of and a rigouros introduction to the theory of marked and unmarked PP’s are referred to the two-volume monograph [140]. Less technical approaches combined with statistics of point processes are presented in [265, 489] and in the survey papers [22, 488]. In the following we reduce the rigour and in particular measurability questions will be not considered.
An advantage of the counting measure approach to point processes (in contrast to modelling with discrete random closed sets) consists in catching random multiplicities of the point or atoms which is important in several fields of application. For example, the description of batch arrivals in queueing systems or of end-points of edges in planar random tessellations requires multiplicities of points. On the other hand, for quite a few point processes in particular for the upmost occurring in stochastic geometry it is natural to assume that at most one point occurs in any \(x \in {\mathbb{R}}^{d}\,\), more precisely
MPP’s \(\Psi _{\mathsf{M}} \sim P_{\mathsf{M}}\) satisfying the later condition are called simple. In view of (4.1) we get \(\mathrm{supp}(\Psi _{\mathsf{M}}) =\{ (X_{i},M_{i}) : i \in \mathbb{N}\}\) for the support set of a simple MPP Ψ M which motivates the somewhat loose writing \(\Psi _{\mathsf{M}} =\{ (X_{i},M_{i}) : i \in \mathbb{N}\}\,\). By the P-a.s. local boundedness of the counting measure Ψ M its support set has no accumulation point in \({\mathbb{R}}^{d}\) and can therefore be considered as a discrete random closed set. The characterization of distributions of random closed sets Ξ by the family of hitting probabilities P(Ξ ∩ K≠∅) for any compact set \(K \subset {\mathbb{R}}^{d}\) leads to the following pendant to Theorem 4.1:
Theorem 4.2.
Let \(\Psi _{\mathsf{M}} \sim P_{\mathsf{M}}\) be a simple MPP on \({\mathbb{R}}^{d}\) with Polish mark space M. Then the distribution P M is completely determined by the void probabilities
for all compact \(X \subset {\mathbb{R}}^{d} \times \mathsf{M}\,\) .
For the sake of simplicity we shall consider only simple MPP’s after the introductory Sect. 4.1.1.
The simplest numerical characteristic of a MPP \(\varPhi _{\mathsf{M}} \sim P_{\mathsf{M}}\) describes the mean number of points in bounded sets \(B \in \mathcal{B}({\mathbb{R}}^{d})\) having marks in an arbitrary set \(L \in \mathcal{B}(\mathsf{M})\). In this way we obtain the intensity measureΛ M (on \(\mathcal{B}({\mathbb{R}}^{d}) \otimes \mathcal{B}(\mathsf{M})\)) defined by
provided that Λ(B) : = Λ M (B ×M) < ∞ for any bounded \(B \in \mathcal{B}({\mathbb{R}}^{d})\) expressing the local finiteness of Λ M . By Theorem 4.1 we are now in a position to define the marked Poisson process \(\varPhi _{\mathsf{M}} \sim \Pi _{\Lambda _{\mathsf{M}}}\) with a given locally finite intensity measure Λ M :
Definition 4.2.
A marked Poisson process \(\varPhi _{\mathsf{M}} \sim \Pi _{\Lambda _{\mathsf{M}}}\) (more precisely its distribution) is completely determined by the following two conditions:
-
1.
\(\varPhi _{\mathsf{M}}(B_{1} \times L_{1}),\ldots ,\varPhi _{\mathsf{M}}(B_{k} \times L_{k})\) are mutually independent random variables for pairwise disjoint \(B_{j} \times L_{j} \in \mathcal{B}({\mathbb{R}}^{d}) \times \mathcal{B}(\mathsf{M})\) with bounded B j for j = 1, …, k and \(k \in \mathbb{N}\).
-
2.
Ψ M (B ×L) is Poisson distributed with mean Λ M (B ×L) for any \(B \times L \in \mathcal{B}({\mathbb{R}}^{d}) \times \mathcal{B}(\mathsf{M})\) with bounded B.
Remark 4.2.
Since Λ M (B ×L) ≤ Λ(B) there exists (by the Radon–Nikodym theorem and disintegration arguments) a family \(\{Q_{\mathsf{M}}^{x},\,x \in {\mathbb{R}}^{d}\}\) of (regular) conditional distributions on \((\mathsf{M},\mathcal{B}(\mathsf{M}))\,\) such that \(\Lambda _{\mathsf{M}}(B \times L) = \int \nolimits \nolimits _{{\mathbb{R}}^{d}}\,Q_{\mathsf{M}}^{x}(L)\,\Lambda (dx)\,\), which justifies the interpretation \(Q_{\mathsf{M}}^{x}(L) = \mathbf{P}(\,M_{i} \in L\,\mid \,X_{i} = x\,)\). It turns out that \(\varPhi _{\mathsf{M}} \sim \Pi _{\Lambda _{\mathsf{M}}}\) can be obtained from an unmarked Poisson processΨ ∼ Π Λ with intensity measureΛ by location-dependent, independent marking, that is, to each atom X i of Ψ located in \(x \in {\mathbb{R}}^{d}\) the mark M i is assigned according to the probability law \(Q_{\mathsf{M}}^{x}\) independent of the location of all other atoms of Ψ and also independent of all other marks even those assigned to further atoms located in x (if Ψ({x}) > 1).
Remark 4.3.
A marked Poisson process \(\varPhi _{\mathsf{M}} \sim \Pi _{\Lambda _{\mathsf{M}}}\) is simple iff the intensity measure Λ( ⋅) is diffuse, i.e. Λ M ({x} ×M) = 0 for all \(x \in {\mathbb{R}}^{d}\), see [346].
Remark 4.4.
The conditions (1) and (2) in the above definition of \(\varPhi _{\mathsf{M}} \sim \Pi _{\Lambda _{\mathsf{M}}}\) can be expressed equivalently by means of the characteristic function of the random vector \((\varPhi _{\mathsf{M}}(B_{1} \times L_{1}),\ldots ,\varPhi _{\mathsf{M}}(B_{k} \times L_{k}))\) as follows: For any \(k \in \mathbb{N}\), any pairwise disjoint \(B_{1} \times L_{1},\ldots ,B_{k} \times L_{k} \in \mathcal{B}({\mathbb{R}}^{d}) \times \mathcal{B}(\mathsf{M})\) and all \(u_{1},\ldots ,u_{k} \in {\mathbb{R}}^{d}\)
Next, we give an elementary explicit construction of an unmarked Poisson process Ψ ∼ Π Λ with locally finite intensity measure Λ( ⋅) , see [102]. This construction is also the basis for simulations of Poisson processes in bounded Borel sets.
Let \(\{K_{m},\,m \in \mathbb{N}\}\) be a partition of \({\mathbb{R}}^{d}\) into bounded Borel sets. Consider an array of independent random variables \(\{\tau _{m},X_{mj}\}_{m,j\in \mathbb{N}}\) defined on \((\Omega ,\mathcal{F},\mathbf{P})\) such that τ m and X mj take values in \(\mathbb{Z}_{+}\) and \({\mathbb{R}}^{d}\) respectively, namely,
-
1.
τ m is Poisson distributed with mean Λ(K m ) (briefly \(\tau _{m} \sim \mathrm{Pois}(\Lambda (K_{m}))\)).
-
2.
\(\mathbf{P}(X_{mj} \in C) = \left \{\begin{array}{@{}l@{\quad }l@{}} \Lambda (C \cap K_{m})/\it\Lambda (K_{m}),\quad &\it\Lambda (K_{m})\neq 0, \\ 0, \quad &\mbox{ otherwise}\end{array} \right.\quad\) for any \(C \in \mathcal{B}({\mathbb{R}}^{d})\,\).
Here Y ∼ Pois(0) means that Y ≡ 0. Clearly, \(\mathbf{P}(X_{mj} \in C) = \mathbf{P}(X_{mj} \in C \cap K_{m})\) for \(m,j \in \mathbb{N}\). Note that the random variables X mj are uniformly distributed on K m if Λ(K m ) is a multiple of \(\nu _{d}(K_{m})\,\).
For any \(B \in \mathcal{B}({\mathbb{R}}^{d})\) and \(m \in \mathbb{N}\) put
Obviously, Ψ m (B) is a random variable for each \(m \in \mathbb{N}\) and any \(B \in \mathcal{B}({\mathbb{R}}^{d})\) such that
Moreover, it turns out that Ψ( ⋅) is a locally finite random counting measure.
Theorem 4.3.
Let Λ be a locally finite measure on \(({\mathbb{R}}^{d},\mathcal{B}({\mathbb{R}}^{d}))\) . For any partition of \(\,\,{\mathbb{R}}^{d}\) into bounded Borel sets K m , \(m \in \mathbb{N}\) , the family of non-negative, integer-valued random variables \(\{\varPhi (B),B \in \mathcal{B}({\mathbb{R}}^{d})\}\) introduced in (4.4) defines an unmarked Poisson process with intensity measure Λ(⋅) .
Proof.
For any \(m,k \in \mathbb{N}\) consider pairwise disjoint Borel sets \(B_{1},\ldots ,B_{k} \subset K_{m}\). Then \(\varPhi (B_{r}) = \varPhi _{m}(B_{r})\) for r = 1, …, k and it follows that
where \(u_{1},\ldots ,u_{k} \in {\mathbb{R}}^{1}\) and \(\varphi _{n}(u_{1},\ldots ,u_{k})\) is the characteristic function of the random vector \(Y _{n} = (Y _{n1},\ldots ,Y _{nk})\) with components \(Y _{nr} =\sum _{ j=1}^{n}\mathbf{1}(X_{mj} \in B_{r})\) for r = 1, …, k.
By setting \(Y _{n0} =\sum _{ j=1}^{n}\mathbf{1}(X_{mj} \in B_{0})\) for \(B_{0} = K_{m} \setminus \cup _{r=1}^{n}B_{r}\) we get a multinomially distributed random vector \((Y _{n0},Y _{n1},\ldots ,Y _{nk})\) with success probabilities \(p_{r} = \mathbf{P}(X_{mj} \in B_{r}) = \it\Lambda (B_{r})/\it\Lambda (K_{m})\), r = 0, 1, …, k , i.e.,
for \(l_{0},l_{1},\ldots ,l_{k} \geq 0\) and \(\sum _{r=0}^{k}l_{r} = n\). Hence, \(\zeta _{n}(u_{0},u_{1},\ldots ,u_{k}) = {(\,p_{0}\,\mathrm{{e}}^{i\,u_{0}} + p_{1}\,\mathrm{{e}}^{i\,u_{1}} + \ldots + p_{k}\,\mathrm{{e}}^{i\,u_{k}}\,)}^{n},u_{0},\ldots ,u_{k} \in {\mathbb{R}}^{1}\,\) is the characteristic function of \((Y _{n0},Y _{n1},\ldots ,Y _{nk})\) and we obtain
Using (4.5) and the latter relation together with \(p_{0} = 1 - p_{1} -\ldots - p_{k}\) we may write
This, however, is just (4.3) (rewritten for the unmarked case) for pairwise disjoint Borel sets \(B_{1},\ldots ,B_{k} \subset K_{m}\,\).
Now, consider pairwise disjoint bounded \(B_{1},\ldots ,B_{k} \in \mathcal{B}({\mathbb{R}}^{d})\) for \(k \in \mathbb{N}\) so that
Obviously, \((\varPhi _{m}(B_{1} \cap K_{m}),\ldots ,\varPhi _{m}(B_{k} \cap K_{m}))_{m\in \mathbb{N}}\) forms a sequence of independent random vectors with independent components as we have proved above. Since \(\varPhi (B_{r}) =\sum _{ m=1}^{\infty }\varPhi _{m}(B_{r} \cap K_{m})\) and the summands are nonnegative, it follows that
implying P( Ψ(B r ) < ∞ ) = 1 for r = 1, …, k, and
Since, for each fixed \(n \in \mathbb{N}\), the sums at the left-hand side of the latter relation are mutually independent, it follows that the limits \(\varPsi (B_{1}),\ldots ,\varPsi (B_{k})\) are independent as well. Finally, in view of the fact that
we conclude that \(\varPsi (B_{r}) \sim \mathrm{Pois}(\it\Lambda (B_{r}))\) for r = 1, …, k, which completes the proof of Theorem 4.3. □
Next, we introduce the translation- and rotation operator T x resp. \(R_{O}\,\mid \mathcal{N}_{\mathsf{M}}\mapsto \mathcal{N}_{\mathsf{M}}\,\) by
for \(\varPsi \in \mathcal{N}_{\mathsf{M}}\,,\,x \in {\mathbb{R}}^{d}\) and O ∈ SO d = group of orthogonal d ×d-matrices with determinant equal to 1. Then the MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) is said to be (strictly) stationary or homogeneous if, for all \(x \in {\mathbb{R}}^{d}\) and bounded Borel sets \(B_{1},\ldots ,B_{k}\),
and isotropic if, for all O ∈ SO d and bounded Borel sets \(B_{1},\ldots ,B_{k}\),
A MPP is said to be motion-invariant if it is both stationary and isotropic.
The stationarity of \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) implies the shift-invariance of the locally finite measure Λ M (( ⋅) ×L) for any fixed \(L \in \mathcal{B}(\mathsf{M})\) provided the intensity
of the unmarked point process Ψ ∼ P exists. This entails that the intensity measure Λ M (B ×L) is a multiple of the Lebesgue-measure ν d (B) which can be rewritten as
where \(Q_{\mathsf{M}}^{o}\) is called the distribution of the typical mark or plainly the mark distribution.
4.1.2 Higher-Order Moment Measures and Palm Distributions
From now on we suppose that the MPP \(\varPsi _{\mathsf{M}} =\sum _{i\geq 1}\delta _{(X_{i},M_{i})} \sim P_{\mathsf{M}}\) is simple. Under the additional assumption \(\mathbf{E}({(\varPsi _{\mathsf{M}}(B \times \mathsf{M}))}^{k}) < \infty \) for some integer k ≥ 2 and any bounded Borel set \(B \subset {\mathbb{R}}^{d}\) we define the factorial moment measure \(\alpha _{\mathsf{M}}^{(k)}\) on \(\mathcal{B}({({\mathbb{R}}^{d} \times \mathsf{M})}^{k})\) by
which is dominated by the kth-order factorial moment measure
of the unmarked simple point process \(\varPsi =\sum _{i\geq 1}\delta _{X_{i}} \sim P\). Note that the sum ∑ ≠ stretches over pairwise distinct indices indicated under the sum sign. If the sum is taken over all k-tuples of indices we get the (ordinary) kth-order moment measure.
For any fixed \(L_{1},\ldots ,L_{k} \in \mathcal{B}(\mathsf{M})\) we obtain as Radon–Nikodym derivative a family of distributions \(Q_{\mathsf{M}}^{x_{1},\ldots ,x_{k}}(L_{1} \times \cdots \times L_{k})\) (for α(k)-almost every \({(x_{1},\ldots ,x_{k})}^{\top }\in {({\mathbb{R}}^{d})}^{k}\)) satisfying
where the integrand is interpreted as (regular) conditional distribution
Definition 4.3.
The stochastic kernel \(Q_{\mathsf{M}}^{x_{1},\ldots ,x_{k}}(A)\) (which is defined to be \(\mathcal{B}({\mathbb{R}}^{dk})\)-measurable in \({(x_{1},\ldots ,x_{k})}^{\top }\in {({\mathbb{R}}^{d})}^{k}\) and a probability measure in \(A \in \mathcal{B}({\mathsf{M}}^{k})\)) is called kth-order or k-point Palm mark distribution of the MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\).
If \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) is stationary then the factorial moment measures \(\alpha _{\mathsf{M}}^{(k)}\) as well as α(k) are invariant under diagonal shifts, i.e.,
which in turn implies \(Q_{\mathsf{M}}^{x_{1}+x,\ldots ,x_{k}+x}(A) = Q_{\mathsf{M}}^{x_{1},\ldots ,x_{k}}(A)\) for all \(x \in {\mathbb{R}}^{d}\,\). By disintegration with respect to the Lebesgue-measure ν d (see [140, Vol. II] for the more details) we can introduce so-called reduced kth-order factorial moment measures \(\alpha _{\mathsf{M},red}^{(k)}\) and \(\alpha _{red}^{(k)}\) by
where λ > 0 stands for the intensity of Ψ ∼ P which already occurred in (4.6). Putting \(\alpha _{red}^{(k)}{\Bigl( \times _{j=2}^{k}\,B_{j}\,\Bigr)} = \alpha _{\mathsf{M},red}^{(k)}{\Bigl(\mathsf{M} \times \times _{j=2}^{k}\,((B_{j} - x) \times \mathsf{M})\,\Bigr)}\) we obtain an analogous relation between α(k) and \(\alpha _{red}^{(k)}\) for the unmarked PP Ψ ∼ P. Rewriting this relation as integrals over indicator functions we are led by algebraic induction to
for any non-negative \(\mathcal{B}({\mathbb{R}}^{dk})\)-measurable function \(f\, :\, {({\mathbb{R}}^{d})}^{k}\mapsto {\mathbb{R}}^{1}\). Setting\(f(x_{1},x_{2},\ldots ,x_{k}) = \mathbf{1}(x_{1} \in B)\,\mathbf{1}((x_{2} - x_{1},\ldots ,x_{k} - x_{1}) \in C)\) for an arbitrary bounded set \(C \in \mathcal{B}({\mathbb{R}}^{d(k-1)})\) and any \(B \in \mathcal{B}({\mathbb{R}}^{d})\) with ν d (B) = 1, for example B = [0, 1]d, we arrive at the formula
which again confirms that \(\alpha _{red}^{(k)}\) is a locally-finite measure on \(({\mathbb{R}}^{d(k-1)},\mathcal{B}({\mathbb{R}}^{d(k-1)}))\). Below it will be shown that \(\alpha _{red}^{(k)}\) coincides with the (k − 1)st-order factorial moment measure with respect to the reduced Palm distribution. It should be mentioned that both factorial moment measures α(k) and \(\alpha _{red}^{(k)}\) are symmetric in their components and the reduction is possible in any component of α(k) yielding \(\alpha _{red}^{(k)}(-C) = \alpha _{ red}^{(k)}(C)\,\). Furthermore, the above reduction and therefore the definition of \(\alpha _{red}^{(k)}\) needs only the diagonal shift-invariance of α(k) (and of the intensity measure Λ) and not the shift-invariance of the finite-dimensional distributions of all orders and even not of the kth order.
This fact gives rise to consider weaker notions of kth-order (moment) stationarity, see Sect. 4.2.1 for k = 2 .
A rather technical, but useful tool in expressing and studying dependences between distant parts of a stationary point pattern is based on so-called (factorial) cumulant measures of both marked and unmarked PP’s. The origins of these characteristics can be traced back up to the beginning of the systematic study of random processes, random fields and particle configurations in statistical mechanics. In probability theory cumulants of random variables or vectors are defined by logarithmic derivatives of the corresponding moment-generating or characteristic functions. Along this line cumulant measures of point processes are defined by mixed higher-order partial derivatives of the logarithm of the probability generating functional of the point process, see [140, Chap. 9]. This approach is the background of the following
Definition 4.4.
For any fixed \(L_{1},\ldots ,L_{k} \in \mathcal{B}(\mathsf{M})\) and bounded \(B_{1},\ldots ,B_{k} \in \mathcal{B}({\mathbb{R}}^{d})\) we define the kth-order factorial cumulant measure \(\gamma _{\mathsf{M}}^{(k)}\) of the MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) by
where the sum \(\sum _{K_{1}\cup \cdots \cup K_{j}=\{1,\ldots ,k\}}\) is taken over all partitions of the set {1, …, k} into j non-empty sets \(K_{1},\ldots ,K_{j}\) and | K r | denotes the cardinality of K r .
In general, \(\gamma _{\mathsf{M}}^{(k)}\) is a locally finite signed measure on \(\mathcal{B}({({\mathbb{R}}^{d} \times \mathsf{M})}^{k})\) which in case of a stationary MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) can also be reduced in analogy to \(\alpha _{\mathsf{M}}^{(k)}\) which leads to
By setting in the latter formula \(L_{1} = \ldots = L_{k} = \mathsf{M}\) we obtain the corresponding relationship between the kth-order factorial cumulant measure γ(k) and the reduced kth-order factorial cumulant measure \(\gamma _{red}^{(k)}\) of the unmarked point process Ψ ∼ P.In the special case k = 2 we have
Finally, if in addition α(k) is absolutely continuous with respect to the Lebesgue measure ν dk then the kth-order product density \({\rho }^{(k)} : {\mathbb{R}}^{d(k-1)}\mapsto [0,\infty ]\) and the kth-order cumulant density \({c}^{(k)} : {\mathbb{R}}^{d(k-1)}\mapsto [-\infty ,\infty ]\) exist such that
The interpretation of ρ(k) as density of k-point subsets of stationary point configurations is as follows:
and in a similar way \({c}^{(2)}(x)\,dx = \mathbf{P}(\varPsi (dx) = 1\mid \varPsi (\{o\}) = 1) -\mathbf{P}(\varPsi (dx) = 1)\) for any x≠o.
Note that in statistical physics \({\rho }^{(k)}(x_{ 1},\ldots ,x_{k-1})\) and \({c}^{(k)}(x_{ 1},\ldots ,x_{k-1})\) are frequently used under the name kth-order correlation function resp. kth-order truncated correlation function.
In some cases under slight additional assumptions the knowledge of the (factorial) moment—or cumulant measures and their densities of any order determine the distribution of point processes uniquely. So far this moment problem for point processes is not completely solved. Another longstanding question which is still unanswered to the best of the author’s knowledge is: Which properties of a locally-finite measure on \(({\mathbb{R}}^{dk},\mathcal{B}({\mathbb{R}}^{dk}))\) are sufficient and necessary for being a kth-order (factorial) moment measure of some unmarked point process Ψ ∼ P?
In the simplest case of a Poisson process with given intensity measure we have the following characterization:
Theorem 4.4.
A MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) with intensity measure Λ M is a marked Poisson process, i.e. \(P_{\mathsf{M}} = \Pi _{\it\Lambda _{\mathsf{M}}}\,\), iff
For stationary unmarked PP’s with intensity λ this means \({\rho }^{(k)}(x) = {\it\Lambda }^{k-1}\) or equivalently c (k)(x) ≡ 0 for all \(x \in {\mathbb{R}}^{d(k-1)}\) and any k ≥ 2 .
If the marks are real-valued it is natural to consider the higher-order mixed moments and mixed cumulants between marks conditional on their locations. For a MPP with mark space \(\mathsf{M} = \mathbb{R}\) and k-point Palm mark distribution \(Q_{\mathsf{M}}^{x_{1},\ldots ,x_{k} }\) let us define
For stationary MPP the function \(c_{mm}(x) = m_{1,1}(o,x)\) for x≠o has been introduced by D. Stoyan in 1984 in order to describe spatial correlations of marks by means of the mark correlation function \(k_{mm}(r) = c_{mm}(x)/{\mu }^{2}\) for \(\|x\| = r > 0\,\), where \(\mu = \int \nolimits \nolimits _{{\mathbb{R}}^{1}}mQ_{\mathsf{M}}^{o}(dm)\) denotes the mean value of the typical mark, see [265, Chap. 5.3], for more details on the use and [438] for a thorough discussion of this function. Kernel-type estimators of the function c mm (x) and their asymptotic properties including consistency and asymptotic normality have been studied in [233] by imposing total variation conditions on higher-order reduced cumulant measures \(\gamma _{\mathsf{M},red}^{(k)}\,\).
Finally, we give a short introduction to general Palm distributions of (marked) point processes. We first consider a simple stationary unmarked point process Ψ ∼ P with positive and finite intensity λ = E Ψ([0, 1]d). Let us define the product measure μ! on \(({\mathbb{R}}^{d} \times \mathcal{N},\mathcal{B}({\mathbb{R}}^{d}) \otimes \mathfrak{N})\) by
for bounded \(B \in \mathcal{B}({\mathbb{R}}^{d})\) and \(A \in \mathfrak{N}\,\), where the exclamation mark indicates that the atom of T x ψ in the origin, i.e., the atom of ψ in \(x \in {\mathbb{R}}^{d}\,\), is removed from each counting measure \(\varPsi \in \mathcal{N}\,\); s(ψ) is shorthand for the support\(\mathrm{supp}(\varPsi ) =\{ x \in {\mathbb{R}}^{d} : \varPsi (\{x\}) > 0\}\).
By the stationarity of Ψ , that is P ∘ T x = P for any \(x \in {\mathbb{R}}^{d}\,\) combined with standard arguments from measure theory it is easily seen that
-
1.
\({\mu }^{!}(\,(B + x) \times A\,) = {\mu }^{!}(\,B \times A\,)\) for any \(x \in {\mathbb{R}}^{d}\).
-
2.
\({P}^{o\,!}(A) := {\mu }^{!}(\,{[0, 1]}^{d} \times A\,)\) for \(A \in \mathfrak{N}\) is a probability measure on \((\mathcal{N}, \mathfrak{N})\,\).
which is concentrated on the subset \({\mathcal{N}}^{o} =\{ \varPsi \in \mathcal{N} : \varPsi (\{o\}) = 0\}\) of counting measures having no atom in the origin o and called the reduced Palm distribution of Ψ ∼ P . As an immediate consequence of (1) and (2) we obtain the factorization
which in turn implies, by algebraic induction, the Campbell–Mecke formula—also known as refined Campbell theorem
for any non-negative \(\mathcal{B}({\mathbb{R}}^{d}) \otimes \mathfrak{N}\)-measurable function \(f\, :\, {\mathbb{R}}^{d} \times \mathcal{N}\mapsto {\mathbb{R}}^{1}\,\). This formula connects the stationary distribution P and the reduced Palm distribution P o ! in a one-to-one correspondence. P o ! (A) can be interpreted (justified in a rigorous sense by limit theorems, see [346]) as probability that Ψ − δ o ∈ A conditional on the null event Ψ({o}) > 0 . Loosely speaking, P o ! describes the stationary point pattern by an observer sitting in a “typical atom” shifted in the origin.
To describe the distributional properties of stationary PP’s it is often more effective to use P o ! rather than P , for example in case of recurrent, semi-Markov- or infinitely divisible PP’s, see for example [346].
A crucial result in this direction is Slivnyak’s characterization of homogeneous Poisson processes:
Theorem 4.5.
A stationary (unmarked) PP Ψ ∼ P on \({\mathbb{R}}^{d}\) with intensity 0 < λ < ∞ is a Poisson process, i.e. P = Π λ iff P = P o ! .
As announced above we apply (4.8) to prove that, for any k ≥ 2 , the kth-order reduced factorial moment measure \(\alpha _{red}^{(k)}\) is nothing else but the (k − 1)st-order factorial moment measure w.r.t the reduced Palm distribution, formally written:
for any non-negative Borel-measurable function f on \({\mathbb{R}}^{d(k-1)}\,\). For notational ease we check this only for k = 2. From (4.7) and the very definition of α(2) we get for bounded \(B,C \in \mathcal{B}({\mathbb{R}}^{d})\) with ν d (B) = 1 that
Quite similarly, we can define reduced Palm distributions \(P_{L}^{o\,!}\) for simple stationary MPP’s with respect to any fixed mark set L ∈ M with Q M o(L) > 0. For this we have to replace λ by \(\it\Lambda _{\mathsf{M}}({[0, 1]}^{d} \times L) = \it\Lambda \,Q_{\mathsf{ M}}^{o}(L)\) which leads to the following extension of (4.8):
for any non-negative, measurable function f on \({\mathbb{R}}^{d} \times \mathcal{N}\), where \(\mathcal{N}_{L}^{o} =\{ \varPsi \in \mathcal{N}_{\mathsf{ M}} : \varPsi (\{o\} \times L) = 0\}\,\).
To include mark sets L and, in particular single marks m, with Q M o-measure zero, we make use of the Radon–Nikodym derivative of the so-called reduced Campbell measureC M ! defined by
with respect to the intensity measure \(\it\Lambda _{\mathsf{M}}(B \times L) = \it\Lambda \,\nu _{d}(B)\,Q_{\mathsf{M}}^{o}(L)\). The corresponding Radon–Nikodym density \(P_{m}^{x\,!}(A)\) is called the reduced Palm distribution of \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) with respect to (x, m) and can be heuristically interpreted as conditional probability that Ψ − δ(x, m) ∈ A given a marked point at x with mark m. This interpretation remains also valid for non-stationary MPP’s and can even be generalized in an appropriate way to k-point reduced Palm distributions \(P_{m_{1},\ldots ,m_{k}}^{x_{1},\ldots ,x_{k}\,!}(A)\) of \(A \in \mathfrak{N}_{\mathsf{M}}\) with respect to \((x_{1},m_{1}),\ldots ,(x_{k},m_{k})\) with pairwise distinct \(x_{1},\ldots ,x_{k}\).
In the stationary case we get \(P_{m_{1},\ldots ,m_{k}}^{x_{1},x_{2}\ldots ,x_{k}\,!}(A) = P_{m_{ 1},\ldots ,m_{j},\ldots ,m_{k}}^{x_{1}-x_{j},\ldots ,o,\ldots ,x_{k}-x_{j}\,!}(T_{ -x_{j}}A)\) for each j = 1, …, k (due to the intrinsic symmetry), which, for k = 1, yields the Campbell–Mecke-type formula
for any non-negative measurable function f on \({\mathbb{R}}^{d} \times \mathsf{M} \times \mathcal{N}_{\mathsf{M}}\,\). Furthermore, this formula can be extended for k ≥ 2 to the following relationship involving the k-point reduced Palm distribution, k-point Palm mark distribution and the k-order reduced factorial moment measure introduced at the beginning of Sect. 4.1.2:
for any non-negative measurable function f k on \({({\mathbb{R}}^{d} \times \mathsf{M})}^{k} \times \mathcal{N}_{\mathsf{M}}\,\).
4.1.3 Different Types of Marking and Some Examples
In the following we distinguish three types of MPP’s \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) by the dependences between marks at distinct points in \({\mathbb{R}}^{d}\), by interactions between separated parts of the point pattern and, last but not least, by cross correlations between the whole point pattern and the whole mark field. In the most general case, only the family of k-point Palm mark distributions seems to be appropriate to describe such complicated structure of dependences.
4.1.3.1 Independently Marked (Stationary) Point Processes
Given an unmarked (not necessarily stationary) PP \(\varPsi =\sum _{i\geq 1}\delta _{X_{i}} \sim P\) on \({\mathbb{R}}^{d}\) and a stochastic kernel \(Q(x,L)\,,\,x \in {\mathbb{R}}^{d},L \in \mathcal{B}(\mathsf{M})\) we assign to an atom X i located at x the mark M i ∼ Q(x, ⋅) independently of \(\varPsi - \delta _{X_{i}}\) and of any other mark M j , j≠i . The resulting MPP \(\varPsi _{\mathsf{M}} =\sum _{i\geq 1}\delta _{(X_{i},M_{i})}\) is said to be derived from Ψ by location-dependent independent marking. We obtain for the intensity measure and the k-point Palm mark distribution
where Λ denotes the intensity measure of Ψ.
Note that the MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) is stationary iff Ψ ∼ P is stationary and independent of an i.i.d. sequence of marks {M i , i ≥ 1} with a common distribution Q( ⋅)—the (mark) distribution of the typical mark M 0 .
4.1.3.2 Geostatistically or Weakly Independently Marked Point Processes
Let unmarked PP \(\varPsi =\sum _{i\geq 1}\delta _{X_{i}} \sim P\) on \({\mathbb{R}}^{d}\) be stochastically independent of a random field \(\{M(x),\,x \in {\mathbb{R}}^{d}\}\) taking values in the measurable mark space \((\mathsf{M},\mathcal{B}(\mathsf{M}))\). To each atom X i we assign the mark \(M_{i} = M(X_{i})\) for i ≥ 1. In this way the k-point Palm mark distribution coincide with the finite-dimensional distributions of the mark field, that is,
Note that the MPP \(\varPsi _{\mathsf{M}} \sim P_{\mathsf{M}}\) is stationary iff both the point process Ψ ∼ P and the random field are (strictly) stationary. In case of real-valued marks (stationary) Gaussian random fields M(x) with some covariance function, see Definition 9.10, or shot-noise fields \(M_{g}(x) =\sum _{i\geq 1}g(x - \xi _{i})\) with some response function \(g\, :\, {\mathbb{R}}^{d}\mapsto {\mathbb{R}}^{1}\) and a (stationary Poisson) point process {ξ i , i ≥ 1} chosen independently of Ψ (see Sect. 9.2.5 for more details) are suitable examples for mark fields.
4.1.3.3 General Dependently Marked Point Processes
In this case the locations X i of the marked atoms and their associated marks M i may depend on each other and, in addition, there are intrinsic interactions within the point field {X i } as well as within the mark field {M i } . This means that the k-point Palm mark distribution \(Q_{\mathsf{M}}^{x_{1},\ldots ,x_{k}}\) must be considered as an conditional distribution, in particular, \(Q_{\mathsf{M}}^{o,x}(L \times \mathsf{M})\) does not coincide with \(Q_{\mathsf{M}}^{o}(L)\).
4.1.3.4 Examples
4.1.3.5 Germ-Grain Processes: Germ-Grain Models
A stationary independently MPP \(\varPsi _{\mathsf{M}} =\{ (X_{i},\Xi _{i})\,,\,i \geq 1\}\) on \({\mathbb{R}}^{d}\) with mark space \(\mathsf{M} = {\mathcal{K}}^{d}\) (= space of all non-empty compact sets in \({\mathbb{R}}^{d}\) equipped with the Hausdorff metric) is called germ-grain process or particle process driven by the PP Ψ = { X i , i ≥ 1} of germs and the typical grainΞ 0 ∼ Q . The associated random set \(\Xi = \bigcup \nolimits _{i\geq 1}(\Xi _{i} + X_{i})\) is called germ-grain model. Note that in general Ξ need not to be closed (P-a.s.). The condition
is sufficient to ensure the P-a.s.-closedness of Ξ , see [229]. The most important and best studied germ-grain model is the Poisson-grain model (also called Boolean model) driven by a Poisson process Ψ ∼ Π Λ of germs {X i , i ≥ 1} , see for example [366, 489] for more details.
4.1.3.6 Poisson-Cluster Processes
If the typical grain \(\Xi _{0} =\{ Y _{1},\ldots ,Y _{N_{0}}\}\) is a P-a.s. finite random point set satisfying (4.11) then the discrete random closed set \(\Xi = \bigcup \nolimits _{i\geq 1}\{Y _{1}^{(i)} + X_{i},\ldots ,Y _{N_{i}}^{(i)} + X_{i}\}\) coincides with the support of a random locally finite counting measure Ψ cl and is called a cluster point process with the PP \(\varPsi _{c} =\{ X_{i},\,i \geq 1\}\) of cluster centres and the typical cluster \(\{Y _{1},\ldots ,Y _{N_{0}}\}\). Factorial moment and cumulant measures of any order can be expressed in terms of the corresponding measures of Ψ and the finite PP \(\sum _{i=1}^{N_{0}}\delta _{Y _{ i}}\), see for example [227]. In case of a (stationary) Poisson cluster centre process Ψ c we get a (stationary) Poisson-cluster process Ψ cl, see Sect. 3.1.4 for more details. In particular, if \(\varPsi \sim \Pi _{\it\it\Lambda _{c}\nu _{d}}\) and the random number N 0 with probability generating function g 0(z) is independent of the i.i.d. sequence of random vectors \(Y _{1},Y _{2},\ldots \) in \({\mathbb{R}}^{d}\) with common density function f we obtain a so-called Neyman–Scott processΨ cl with intensity \(\it\it\Lambda _{\mathrm{cl}} = \it\it\Lambda _{c}\,\mathbf{E}N_{0}\,\), second-order product density \({\rho }^{(2)}(x) = {c}^{(2)}(x) + \it\it\Lambda _{\mathrm{cl}}\) and its kth-order cumulant density for k ≥ 2 takes on the form
Compare Definition 3.11 for its special case.
4.1.3.7 Doubly Stochastic Poisson Processes
Now, let Λ be a (stationary) random measure on \({\mathbb{R}}^{d}\), see for example [140] for details. The new unmarked PP \(\varPsi _{\it\it\Lambda } \sim P_{\it\it\Lambda }\) defined by the finite-dimensional distributions
for any disjoint bounded \(B_{1},\ldots ,B_{k} \in \mathcal{B}_{0}({\mathbb{R}}^{d})\) and any \(n_{1},\ldots ,n_{k} \in \mathbb{N} \cup \{ 0\}\), is called doubly stochastic Poisson (or Cox) process with driving measure Λ( ⋅), compare Definition 3.7. In the special case Λ( ⋅) = λ ν d (( ⋅) ∩ Ξ) , where Ξ is a (stationary) random closed set, for example a Boolean model, the (stationary) PP Ψ Λ (called interrupted Poisson process) is considered as a Poisson process restricted on the (hidden) realizations of Ξ. The factorial moment and cumulant measures of Ψ Λ are expressible in terms of the corresponding measures of random driving measure Λ , see for example [289].
4.2 Point Process Statistics in Large Domains
Statistics of stationary point processes is mostly based on a single observation of some point pattern in a sufficiently large domain which is assumed to extend unboundedly in all directions. We demonstrate this concept of asymptotic spatial statistics for several second-order characteristics of point processes including different types of K-functions, product densities and the pair correlation function. Variants of Brillinger-type mixing are considered to obtain consistency and asymptotic normality of the estimators.
The philosophy of large-domain spatial statistics is as follows: Let there be given a single realization of a random point pattern or a more general random set in a sufficiently large sampling window \(W_{n} \subset {\mathbb{R}}^{d}\), which is thought to expand in all directions as n → ∞ . Further, we assume that there is an adequate model describing the spatial random structure whose distribution is at least shift-invariant (stationary) and sometimes additionally even isotropic. Then only using the information drawn from the available observation in W n we define empirical counterparts (estimators) of those parameters and non-parametric characteristics which reflect essential properties of our model. To study the asymptotic behaviour of the estimators such as weak or strong consistency and the existence of limit distributions (after suitable centering and scaling) we let W n increase unboundedly which requires additional weak dependence conditions. Throughout we assume that \(\{W_{n},n \in \mathbb{N}\}\) is a convex averaging sequence, that is,
-
1.
W n is bounded, compact, convex and \(W_{n} \subseteq W_{n+1}\,\) for \(n \in \mathbb{N}\).
-
2.
\(r(W_{n}) :=\sup \{ r > 0 : B_{r}(x) \subseteq W_{n}\:\:\mbox{ for some}\:\:x \in W_{n}\} \uparrow \infty \,\).
The second property means that W n expands unboundedly in all directions and is equivalent to \(\nu _{d-1}(\partial W_{n})/\nu _{d}(W_{n})\mathop\rightarrow\limits _{n\rightarrow \infty }0\) as immediate consequence of the geometric inequality
see [237].
Exercise 4.1.
Show that
for 0 ≤ r ≤ r(W n ) from which, together with \(\nu _{d}(W_{n} \ominus B_{r(W_{n})}(o)) = 0\), the l.h.s. of (4.13) immediately follows. The r.h.s. of (4.13) results from an inequality by J.M. Wills, see [519].
From the mathematical view point it is sometimes more convenient to consider rectangles \(W_{n} = \times _{i=1}^{d}[0,a_{i}^{(n)}]\) with \(a_{i}^{(n)} \uparrow \infty \) for i = 1, …, d or blown up sets W n = n W , where \(W \subset {\mathbb{R}}^{d}\) is a fixed convex body containing the origin o as inner point.
4.2.1 Empirical K-Functions and Other Summary Statistics of Stationary PP’s
Second-order statistical analysis of spatial point patterns is perhaps the most important branch in point process statistics comparable with the spectral density estimation in time series analysis. We assume that the simple unmarked PP \(\varPsi =\sum _{i\geq 1}\delta _{X_{i}}\) has finite second moments, i.e. E Ψ 2(B) < ∞ for all bounded \(B \in \mathcal{B}({\mathbb{R}}^{d})\), and is strictly or at least weakly stationary.
Weak (or second-order) stationarity of an unmarked PP Ψ ∼ P requires only the shift-invariance of the first- and second-order moment measures, i.e. \(\it\it\Lambda (B_{1} + x) = \it\it\Lambda (B_{1})\) and \({\alpha }^{(2)}((B_{1} + x) \times (B_{2} + x)) = {\alpha }^{(2)}(B_{1} \times B_{2})\) for any bounded \(B_{1},B_{2} \in \mathcal{B}({\mathbb{R}}^{d}))\) and all \(x \in {\mathbb{R}}^{d}\). Obviously, strictly stationary point processes having finite second moments are weakly stationary. Further note that the reduced second factorial moment measure \(\alpha _{red}^{(2)}(\cdot )\) is well-defined also under weak stationarity but it can not be expressed as first-order moment measure w.r.t. P o ! as in (4.9), see [140] for more details. In what follows we assume strict stationarity. By applying the Palm and reduction machinery sketched in Sect. 4.1.2 we can describe the first and second moment properties by the intensity λ = E Ψ([0, 1]d) and the reduced second factorial moment measure \(\alpha _{red}^{(2)}(\cdot )\) defined by (4.7) for k = 2 resp. (4.9) as first moment measure with respect to the Palm distribution P o! in case of strict stationarity. If Ψ is additionally strictly or at least weakly isotropic, i.e. \(R_{O}\alpha _{red}^{(2)} = \alpha _{red}^{(2)}\) for O ∈ SO d , then it suffices to know the function \(\alpha _{red}^{(2)}(B_{r}(o))\) for r ≥ 0 . In [424] B. Ripley introduced the K-function
for r ≥ 0 as basic summary characteristic for the second-order analysis of motion-invariant PP’s, see also [22] or [265, Chap. 4.3] for more details and historical background. From (4.9) we see that λ K(r) coincides with conditional expectation E(Ψ{(B r ({o}) ∖ { o})}∣Ψ({o}) = 1 ) giving the mean number of points within the Euclidean distance r from the typical point (which is not counted). If Ψ is a homogeneous Poisson process with intensity λ , then, by Slivnyak’s theorem (see Theorem 4.5 in Sect. 4.1.2), \(\alpha _{red}^{(2)}(\cdot ) = \mathbf{E}\varPsi (\cdot ) = \it\lambda \,\nu _{d}(\cdot )\) and hence we get
For better visualization of the Poisson property by a linear function the so-called L-function \(L(r) :={\Bigl(\, K(r)/\omega {_{d}\,\Bigr)}}^{1/d}\) is sometimes preferred instead of the K-function. Both the K- and L-function represent the same information, but they cannot completely characterize the distribution of a (motion-invariant) PP. In other words, there are different point processes having the same K-function. Further note that an explicit description of the family of K-functions does not exist so far. Nevertheless, the K-function and its empirical variants, see below, are used to check point process hypotheses when the K-function of the null hypothesis is known (or generated by simulation on a finite interval [0, r 0] ), see Sect. 3.3.3 and Figs. 3.47–3.48.
In particular, the simple parabola-shape of the K-function (4.15) facilitates to check the property of complete spatial randomness (briefly CSR) of a given point pattern. Lemma 3.12 shows the connection between CSR and the Poisson property shows the following
It contains the interpretation of the Poisson point process in statistical mechanics as particle configuration, for example molecules in “ideal gases”, modelled as grand canonical ensemble, where neither attraction nor repulsion forces between particles occur. Lemma 3.12 also reveals an easy way to simulate homogeneous Poisson processes in bounded domains, see Algorithm 3.6 in Sect. 3.2.2.
Since the K-function is also used to analyze (second-order) stationary, non-isotropic PP’s we introduce two generalized versions of Ripley’s K-function (4.14). First, the Euclidean d-ball B r (o) in (4.14) is replaced by r B , where \(B \subset {\mathbb{R}}^{d}\) is a compact, convex, centrally symmetric set containing o as inner point. Such set B is called structuring element in image analysis and coincides with the unit ball \(\{x \in {\mathbb{R}}^{d} :\mathrm{ N}_{B}(x) \leq 1\}\) generated by a unique norm N B ( ⋅) on \({\mathbb{R}}^{d}\,\). Let K B (r) denote the analogue to (4.14) which equals \(\nu _{d}(B)\,{r}^{d}\) if \(\varPsi \sim \Pi _{\it\Lambda \nu _{d}}\). In case of a Neyman–Scott process we obtain from (4.12) that
A second generalization of (4.14) is the multiparameter K-function, see [231], defined by
which contains the same information as the centrally symmetric measure \(\alpha _{red}^{(2)}(\cdot )\,\). For stationary Poisson processes we get
We next define three slightly different non-parametric estimators of the function \({\it\Lambda }^{2}\,K_{B}(r)\) (briefly called empirical K-functions):
Each of these empirical processes is non-decreasing, right-continuous, and piecewise constant with jumps of magnitude \(1/\nu _{d}(W_{n})\) (for i = 1, 2) at random positions \(\mathrm{N}_{B}(X_{j} - X_{i})\) arranged in order of size for i≠j. Quite analogously, by substituting the indicators of the events \(\{\mathrm{N}_{B}(X_{j} - X_{i}) \in [0,r]\}\) by the indicators of \(\{X_{j} - X_{i} \in \times _{k=1}^{d}[-r_{k},r_{k}]\}\) we obtain the multivariate empirical processes \({\Bigl(\widehat{{\it\Lambda }^{2}K}\Bigr)}_{n,i}(r_{1},\ldots ,r_{d})\) for i = 1, 2, 3 as empirical counterparts of \({\it\Lambda }^{2}\,K(r_{1},\ldots ,r_{d})\). By (4.8) resp. (4.9), \({\Bigl(\widehat{{\it\Lambda }^{2}K_{B}}\Bigr)}_{n,1}(r)\) is easily seen to be an unbiased estimator for \({\it\Lambda }^{2}\,K_{B}(r)\) but it ignores the edge effect problem, that is, we need information from the dilated sampling window \(W_{n} \oplus r_{0}\,B\) to calculate this estimator for 0 ≤ r ≤ r 0 . If this information is not available then one has to reduce the original window to the eroded set \(W_{n} \ominus r_{0}\,B\,\) which is known as minus sampling. The second estimator needs only the positions of points within W n , however, its bias disappears only asymptotically, i.e.
Finally, \({\Bigl(\widehat{{\it\Lambda }^{2}K_{B}}\Bigr)}_{n,3}(r)\) is a so-called edge-corrected or Horvitz–Thompson-type estimator which also needs only the points located within W n . The pairs \((X_{i},X_{j}) \in W_{n} \times W_{n}\) are weighted according to the length and direction of the difference vector \(X_{j} - X_{i}\) providing the unbiasedness of the estimator \(\mathbf{E}{\Bigl(\widehat{{\it\Lambda }^{2}K_{B}}\Bigr)}_{n,3}(r)\).
Exercise 4.2.
Show (4.16) by applying the inequality (4.13) and prove
by means of (4.8) resp. (4.9).
For further details and more sophisticated edge corrections we refer to [265, 385, 489] and references therein.
Before regarding consistency properties of the empirical K-functions we have a short look at the estimation of the simplest summary characteristic—the intensity λ—and its powers λk given by
for any fixed integer k ≥ 2 . A simple application of the Campbell formula (4.6) (or (4.8)) and the definition of the kth-order factorial moment measure yields
which shows the unbiasedness of \(\widehat{\it\lambda }_{n}\) for any stationary PP, whereas \((\widehat{{\it\lambda }^{k}})_{n}\) for k ≥ 2 is unbiased only for the Poisson process \(\varPsi \sim \Pi _{\it\Lambda \,\nu _{d}}\).
Exercise 4.3.
For a stationary Poisson process \(\varPsi \sim \Pi _{\it\Lambda \,\nu _{d}}\) show that
for any (not necessarily disjoint) bounded sets \(B_{1},\ldots ,B_{k} \in \mathcal{B}({\mathbb{R}}^{d})\).
The decomposition \({\alpha }^{(2)}(W_{n} \times W_{n}) = {\gamma }^{(2)}(W_{n} \times W_{n}) + {\it\lambda }^{2}\,\nu _{d}^{2}(W_{n})\) and reduction reveal the asymptotic unbiasedness of \((\widehat{{\it\lambda }^{2}})_{n}\)
provided that the total variation \(\|\,\gamma _{red}^{(2)}\,\|_{\mathrm{TV}}\) is finite. This motivates the assumption of bounded total variation of the reduced factorial cumulant measure \(\gamma _{red}^{(k)}(\cdot )\) for some k ≥ 2 to express short-range correlation of the point process. To be precise, we rewrite the locally finite (in general not finite) signed measure \(\gamma _{red}^{(k)}(\cdot )\) on \({\Bigl({\mathbb{R}}^{d(k-1)},\mathcal{B}({\mathbb{R}}^{d(k-1)})\Bigr)}\) as difference of the positive and negative part \(\gamma _{red}^{(k)\,+}(\cdot )\) resp. \(\gamma _{red}^{(k)\,-}(\cdot )\) (Jordan decomposition) and define the corresponding total variation measure \({\Bigl|\,\gamma _{red}^{(k)}\,\Bigr|}(\cdot )\) as a sum of the positive and negative part:
Note that the locally finite measures \(\gamma _{red}^{(k)\,+}(\cdot )\) and \(\gamma _{red}^{(k)\,-}(\cdot )\) are concentrated on two disjoint Borel sets H + resp. H − with \({H}^{+} \cup {H}^{-} = {\mathbb{R}}^{d(k-1)}\) (Hahn decomposition) which leads to the total variation of \(\gamma _{red}^{(k)}(\cdot )\;\):
where \({c}^{(k)}\, :\, {\mathbb{R}}^{d(k-1)}\mapsto [-\infty ,\infty ]\) is the kth-order cumulant density, if it exists.
Definition 4.5.
A stationary PP Ψ ∼ P on \({\mathbb{R}}^{d}\) satisfying \(\mathbf{E}{\varPsi }^{k}({[0,1]}^{d}) < \infty \) for some integer k ≥ 2 is said to be B k -mixing if \(\|\,\gamma _{red}^{(j)}\,\|_{\mathrm{TV}} < \infty \) for j = 2, …, k. A B ∞ -mixing stationary PP is called Brillinger-mixing or briefly B-mixing.
Example 4.1.
From (4.12) it is easily seen that a Neyman–Scott process is B k -mixing iff \(\mathbf{E}N_{0}^{k} < \infty \) without restrictions on f. This remains true for any Poisson-cluster process. Moreover, a general cluster process is B k -mixing if the PP Ψ c of cluster centres is B k -mixing and the typical cluster size N 0 has a finite kth moment, see [227] for details and further examples like Cox processes.
Proposition 4.1.
For any B k-mixing stationary PP we have \(\mathbf{E}(\widehat{{\it\lambda }^{k}})_{n}\;\mathop\rightarrow _{n\rightarrow \infty }\;{\it\Lambda }^{k}\) fork ≥ 2 .
We next state the mean-square consistency of the above-defined empirical K-functions under mild conditions. Furthermore, it can be shown that a possible weak Gaussian limit (after centering with mean and scaling with \(\sqrt{ \nu _{d}(W_{n})}\)) is for each of the estimators of \({\it\lambda }^{2}\,K_{B}(r)\) the same.
Theorem 4.6.
Let Ψ ∼ P be a B 4-mixing stationary PP with intensity λ. Then
In other words, the boundary effects are asymptotically neglectable which can be considered as a general rule of thumb in large domain statistics.
Finally, we mention that also higher-order reduced moment measures can be estimated in quite the same way, see for example [274, 289]. Further second-order summary characteristics and their empirical counterparts (called summary statistics) such as the second-order product density ρ(2)(x), the pair correlation functiong(r) and the asymptotic variance \({\sigma }^{2} :=\lim _{n\rightarrow \infty }\nu _{d}(W_{n})\,\mathbf{E}{(\widehat{\it\lambda }_{n} - \it\lambda )}^{2}\,\), see (4.17), are briefly discussed in Sect. 4.2.3. Summary statistics are used in all branches of statistics to summarize data sets—in our case data from point patterns or from realizations of random sets—to describe the underlying models by a small number of parametric and non-parametric estimates. Further summary characteristics frequently used in point process statistics are the empty space function (or contact distribution function) F, the nearest-neighbour distance functionG and the J-function defined for a stationary PP \(\varPsi =\sum _{i\geq 1}\delta _{X_{i}} \sim P\) by
F is the distribution function of the distance dist(x, Ψ) from a fixed point \(x \in {\mathbb{R}}^{d}\) to the nearest atom of Ψ , whereas G is the distribution function of the corresponding distance from a typical atom of Ψ to the nearest other atom of Ψ. Unbiased non-parametric estimators of F(r) and λ G(r) are
The empirical J-function \(\widehat{J}_{n}(r)\) is defined as ratio \(\widehat{\it\lambda }_{n}\,\widehat{F}_{n}(r)/(\,\widehat{\it\lambda \,G}\,)_{n}(r)\,\). To avoid boundary effects we replace W n by \(W_{n} \ominus B_{r}(o)\) for 0 ≤ r ≤ diam(W n ) ∕ 2 , if the point pattern is observable only inside W n . In case of \(\varPsi \sim \Pi _{\it\lambda \nu _{d}}\) Slivnyak’s theorem yields \(F(r) = G(r) = 1 -\exp \{-\it\lambda \,\omega _{d}\,{r}^{d}\}\) so that J(r) ≡ 1. This fact can be used for testing CSR just by regarding the plot of the empirical version \(\widehat{J}_{n}(r)\) in some interval [0, r 0] .
4.2.2 The Role of Ergodicity and Mixing in Point Process Statistics
The assumption of (strict) stationarity of a point process or random closed set under consideration is frequently accompanied by the requirement of ergodicity. It is beyond the scope of this survey to capture the full depth of this notion. We only say that ergodicity is always connected with a group of measure preserving transformations acting on the probability space. In our situation we take quite naturally the group of translations \(\{T_{x} : x \in {\mathbb{R}}^{d}\}\) as defined in Sect. 4.1.1 on the space of (marked) locally-finite counting measures or the corresponding shifts on the space of closed sets in \({\mathbb{R}}^{d}\). To be precise, we define besides ergodicity also the somewhat stronger condition of mixing for stationary (unmarked) PP’s:
Definition 4.6.
A (strictly) stationary PP Ψ ∼ P is said to be ergodic resp. mixing if
for any \(Y _{1},Y _{2} \in \mathfrak{N}\).
Loosely speaking, mixing means that two events becomes nearly independent when they occur over parts of \({\mathbb{R}}^{d}\) being separated by a great distance and ergodicity weakens this distributional property in the sense of Cesaro limits. In physics and engineering one says that an ergodic stochastic process allows to detect its distribution after very long time of observation which carries over to spatial ergodic processes when the observation window expands unboundedly in all directions. This interpretation is rigorously formulated by ergodic theorems which state the P-a.s. convergence of spatial means to expectations with respect to the underlying distribution. The following ergodic theorem by X.X. Nguyen and H. Zessin [382] is of particular importance in the theory as well as in statistics of stationary PP’s.
Theorem 4.7.
Let Ψ ∼ P be a stationary ergodic PP on \({\mathbb{R}}^{d}\) with intensity λ , and let \(g\, :\, \mathcal{N}\mapsto [0,\infty ]\) be \((\mathfrak{N},\mathcal{B}({\mathbb{R}}^{d}))\) -measurable such that \(\int \nolimits \nolimits _{{\mathcal{N}}^{o}}g(\varPsi ){P}^{o\,!}(d\varPsi ) < \infty \,\) . Then
for P-almost every \(\varPsi \in \mathcal{N}\,\) .
This result can be applied to prove strong consistency for many estimators, in particular, for various empirical Palm characteristics. In the special cases (a) g(ψ) ≡ 1 and (b) g(ψ) = ψ(r B) we obtain strong consistency of the intensity estimator (4.17) and \({\Bigl(\widehat{{\it\lambda }^{2}K_{B}}\Bigr)}_{n,1}(r)\) for any r ≥ 0 , which implies even uniformly strong consistency:
We mention just one asymptotic relationship which requires mixing instead of ergodicity, namely the generalized version of Blackwell’s renewal theorem. If the stationary second-order PP Ψ ∼ P is mixing, then, for any bounded \(B \in \mathcal{B}({\mathbb{R}}^{d})\) satisfying ν d (∂B) = 0 , it holds
see [140]. Note that a renewal process is just mixing if the length of the typical renewal interval has a non-arithmetic distribution and thus, the latter result (applied to an bounded interval B = [a, b]) contains the mentioned classical result from renewal theory. For related results concerning the weak convergence of the shifted Palm distribution \({P}^{o\,!}(T_{x}(\cdot ))\) to the stationary distribution P( ⋅) as \(\|x\| \rightarrow \infty \) we refer the reader to [140, 346].
4.2.3 Kernel-Type Estimators for Product Densities and the Asymptotic Variance of Stationary Point Processes
The Lebesgue density ρ(2)(x) of \(\alpha _{red}^{(2)}(\cdot )\)—introduced in Sect. 4.1.2 as second-order product density—and, if Ψ ∼ P is motion-invariant, the pair correlation functiong(r) defined by
are very popular second-order characteristics besides the cumulative K-function. Note that g(r) is understood as derivative (existing for ν1-almost every r ≥ 0) of an absolutely continuous K-function (4.14). Since the numerical differentiation of the empirical versions of K(r) as well as of the multiparameter K-function \(K(r_{1},\ldots ,r_{d})\) leads to density estimators of minor quality, the most statisticians prefer the established method of kernel estimation in analogy to probability density functions. The corresponding edge-corrected kernel estimators for λ ρ(2)(x) and λ2 g(r) are
where the kernel function \(k_{d}\mid {\mathbb{R}}^{d}\mapsto \mathbb{R}\) is integrable (and mostly symmetric, bounded with bounded support) such that \(\int \nolimits \nolimits _{{\mathbb{R}}^{d}}\,k_{d}(x)\,dx = 1\) and the sequence of bandwidths is chosen such that b n ↓ 0 and \(b_{n}^{d}\,\nu _{d}(W_{n})\mathop\rightarrow\limits _{n\rightarrow \infty }\infty \). These conditions imply the pointwise asymptotic unbiasedness of the kernel estimators, namely
at any continuity point x≠o of ρ(2) resp. at any continuity point r > 0 of g , see e.g. [232–234, 275]. Under some further additional conditions one can show that
and also central limit theorems (briefly CLT’s) and optimal bandwidths can be derived, see for example [232] for an application to testing point process models. Furthermore, various asymptotic results for higher-order kernel-type product density estimators (among them rates of convergence, P-a.s. convergence) have been obtained under stronger mixing assumptions, see [234, 275].
Finally, we regard a kernel-type estimator of the limit
which exists for all B 2-mixing stationary PP’s. The following estimator has been studied in [238]:
where \(c_{n} := b_{n}\,{(\nu _{d}(W_{n}))}^{1/d}\) and \(w : {\mathbb{R}}^{d}\mapsto {\mathbb{R}}^{1}\) is a non-negative, symmetric, bounded function with bounded support satisfying lim x → o w(x) = w(o) = 1 .
Theorem 4.8.
For a B 4-mixing stationary PP the estimator \((\widehat{{\sigma }^{2}})_{n}\) is asymptotically unbiased and mean-square consistent if \(b_{n}\mathop\rightarrow\limits _{n\rightarrow \infty }0\,\), \(c_{n}/r(W_{n})\mathop\rightarrow\limits _{n\rightarrow \infty }0\,\), \(c_{n}\mathop\rightarrow\limits _{n\rightarrow \infty }\infty \,\), and \(b_{n}\,c_{n}\mathop\rightarrow\limits _{n\rightarrow \infty }0\,\). If the PP is even B-mixing, then \(\sqrt{\nu _{d } (W_{n } )}\,{\Bigl(\,\widehat{\it\lambda }_{n} - \it\lambda \,\Bigr)}/\sigma \) is asymptotically N(0,1)-distributed, where σ can be replaced by the square root of \((\widehat{{\sigma }^{2}})_{n}\,\) .
In this way one can construct an asymptotic confidence interval which covers the intensity λ with given probability 1 − α .
4.3 Mixing and m-Dependence in Random Point Processes
Large domain statistics requires weak dependence assumptions of the observed spatial process to derive properties of the estimators and to construct asymptotic tests for checking statistical hypotheses. We formulate and apply a spatial ergodic theorem. The notion of m-dependence plays an important role to prove limits theorems for Poisson-driven models demonstrated in particular for the Boolean model and statistics taken from Poisson procesess. We consider also some examples which exhibit appropriate spatial versions of the α- and β-mixing condition.
4.3.1 Poisson-Based Spatial Processes and m-Dependence
Definition 4.7.
A family of random variables \(\{\xi (t),\,t \in {\mathbb{Z}}^{d}\}\) defined on \((\Omega ,\mathcal{F},\mathbf{P})\) is called m-dependent (d-dimensional) random field for some \(m \in \mathbb{N}\) if for any finite \(U,V \subset {\mathbb{Z}}^{d}\) the random vectors (ξ(u)) u ∈ U and (ξ(v)) v ∈ V are independent whenever \(\max \limits _{1\leq i\leq d}\,\vert \,u_{i} - v_{i}\,\vert \, >\, m\) for all \(u = {(u_{1},\ldots ,u_{d})}^{\top }\in U\) and \(v = {(v_{1},\ldots ,v_{d})}^{\top }\in V\), see also Sect. 10.1.2.
For d = 1 we use the term “sequence” instead of “field” and in what follows we shall fix the dimension d ≥ 1 . In particular, in the theory of limit theorems for sums of random fields the particular case of m-dependent random variables indexed by a subset of \({\mathbb{Z}}^{d}\) plays an important role because most of the classical limit theorems known for sums of independent random variables remain valid with obvious modifications for m-dependent sequences and fields. This includes also a number of refined results such as Berry–Esseen bounds and asymptotic expansions of the remainder term in the CLT, see [226], or Donsker’s invariance principle and functional CLT’s for empirical m-dependent processes with càdlàg-trajectories, see for example [69].
In stochastic geometry and point process statistics, m-dependent random fields appear in connection with models which are defined by independently marked Poisson processes. We discuss here two examples which exhibit the main idea. This approach has been successfully applied to derive CLT’s for functionals of Poisson-cluster processes and Poisson-grain models, see for example [227, 236]. For notational ease, let \(W_{n} = \times _{i=1}^{d}[0,a_{i}^{(n)})\) be a rectangle with large enough edges \(a_{1}^{(n)},\ldots ,a_{d}^{(n)}\,\).
Example 4.2.
Let \(\Xi = \bigcup \nolimits _{i\geq 1}(\,\Xi _{i} + X_{i}\,)\) be a Boolean model generated by the stationary Poisson process \(\varPsi \sim \Pi _{\it\lambda \,\nu _{d}}\) and a bounded typical grain satisfying \(\Xi _{0} \subseteq {[-r,r]}^{d}\) P-a.s. for some fixed r > 0 . We are interested in the asymptotic behaviour of the random d-volume \(S_{n} = \nu _{d}(\Xi \cap W_{n})\) which is closely connected with the empirical volume fraction \(\widehat{p}_{n} = S_{n}/\nu _{d}(W_{n})\).
Example 4.3.
We consider the random sum
which coincides up to the scaling factor \(1/\nu _{d}(W_{n})\) with the empirical K-function \({\Bigl(\widehat{{\it\lambda }^{2}K_{B}}\Bigr)}_{n,1}(r)\). We are able to derive the Gaussian limit distribution using the CLT for m-dependent field provided that \(\varPsi \sim \Pi _{\it\lambda \,\nu _{d}}\). For simplicity assume that B ⊆ [ − 1, 1]d .
In both cases take the smallest number r i ≥ r such that the ratio \(v_{i}^{(n)} = a_{i}^{(n)}/2r_{i}\) is an integer for i = 1, …, d and decompose W n into blocks E t with \(t = {(t_{1},\ldots ,t_{d})}^{\top }\) as follows:
Then we may write \(S_{n} =\sum \limits _{t\in V _{n}}\,\xi (t)\) and \(S_{n}(r) =\sum \limits _{t\in V _{n}}\,\xi _{r}(t)\) with the random variables
forming a stationary 1-dependent random field due to the independence properties of the stationary Poisson process Ψ and the fact that grains {Ξ i , i ≥ 1} are i.i.d. and independent of Ψ. By the same arguments we get an i.i.d. sequence of random marked counting measures
and, in addition, the ξ(t)’s admit a representation ξ(t) = f(Ψ y , | t − y | ≤ 1) in terms of a measurable function \(f\, :\, {(\mathcal{N}_{\mathsf{M}}^{0})}^{{3}^{d} }\mapsto {\mathbb{R}}^{1}\,\), where \(\mathcal{N}_{\mathsf{M}}^{0}\) denotes the space of locally-finite marked counting measures on \(\times _{i=1}^{d}[0,2r_{i}) \times \mathcal{K}\). In this way {ξ(t), t ∈ V n } becomes a two-dependent random field with block representation, see [199, 226] for details. This representation of the field by functions of finite blocks of independent random elements allows to check simple conditions that imply explicit bounds of the remainder terms of asymptotic expansions in the CLT for S n and S n (r) as well.
The CLT for (stationary) m-dependent random fields, see for example [69] or references in [226], combined with \(\vert V _{n}\vert = \nu _{d}(W_{n})/{(2\,r)}^{d}\) and \(p = \mathbf{E}\nu _{d}(\Xi \cap [0,1{)}^{d})\) yields
If the compact typical grain Ξ 0 is not strictly bounded, then we first replace Ξ 0 by the truncated grain \(\Xi _{0} \cap {[-r,r]}^{d}\) and apply the above CLT to the corresponding truncated Boolean model Ξ(r). In a second step we show that the ratio
becomes arbitrarily small uniformly in \(n \in \mathbb{N}\) as r grows large provided that \(\mathbf{E}\nu _{d}^{2}(\Xi _{0}) < \infty \,\). Finally, Slutsky’s theorem completes the proof of the CLT in the general case.
In Example 4.3 we immediately obtain the normal convergence
with \(\sigma _{B}^{2}(r) = 2\,\it\lambda \,\nu _{d}(B)\,{r}^{d}\,{\Bigl(1 + 2\,\it\lambda \,\nu _{d}(B)\,{r}^{d}\,\Bigr)}\,\), see also [274] for related CLT’s for B-mixing stationary PP’s. Using the block representation of the random variables ξ r (t), t ∈ V n , and the some results in [226], see also references therein, we obtain the optimal Berry–Esseen bound
where \(\Phi (x) := \mathbf{P}(N(0,1) \leq x)\,,x \in {\mathbb{R}}^{1}\,\), denotes the standard normal distribution function.
Moreover, for the random sum
which equals twice the number of pairs of points having N B -distance less than or equal to r, a local CLT with asymptotic expansion can be proved by methods developed in [199, 226]:
for any k = 0, 1, 2, … , where \(x_{n}(k,r) ={\Bigl(\, 2\,k -\mathbf{E}\widehat{S}_{n}(r)\,\Bigr)}/{\Bigl(\,\mathbf{v}\mathbf{a}\mathbf{r}\,\widehat{S}_{n}{(r)\,\Bigr)}}^{1/2}\) and
4.3.2 Strong Mixing and Absolute Regularity for Spatial Processes
The quantitative assessment of (weak) dependence between parts of spatial processes (e.g. random fields, point processes, random closed sets) over disjoint subsets of \({\mathbb{R}}^{d}\) is based on mixing coefficients. These quantities provide uniform bounds of the dependence between σ-algebras generated by the spatial process over these disjoint set which include rates of decay when the distance between these subsets increases. These mixing coefficients permit to derive covariance estimates of the random variables measurable with respect to these σ-algebras. This in turn is essential in proving asymptotic normality for sums of these random fields defined over \((\Omega ,\mathcal{F},\mathbf{P})\,\). Here we shall briefly discuss two of the most relevant mixing coefficients, see also Sect. 10.1.2.
Definition 4.8.
For any two sub-σ-algebras \(\mathcal{A},\,\mathcal{B}\subset \mathcal{F}\,\) the α-mixing (or strong) coefficient \(\alpha (\mathcal{A},\,\mathcal{B})\) and the β-mixing (or absolute regularity) coefficient \(\beta (\mathcal{A},\,\mathcal{B})\) are defined by
where \(\mathcal{A}\otimes \mathcal{B}\) is the product σ-algebra generated by \(\mathcal{A}\) and \(\mathcal{B}\) and \(\mathbf{P}_{\mathcal{A}}\times \mathbf{P}_{\mathcal{B}}\) denotes the product measure of the corresponding marginal distributions.
The inequality \(2\,\alpha (\mathcal{A},\mathcal{B}) \leq \beta (\mathcal{A},\mathcal{B})\) is immediately seen from the above definition, see [83] for an all-embracing discussion of mixing coefficients. As already mentioned the covariance cov(ξ, η) can be bounded by means of these mixing coefficient with respect to the σ-algebras \(\mathcal{A} = \sigma (\xi )\) and \(\mathcal{B} = \sigma (\eta )\) generated by the random variables ξ and η, respectively. Such covariance bounds are known for long time and can be found in many papers and textbooks on limit theorems for sums of weakly dependent random variables. If ξ, η are real-valued and p , q ∈ [1, ∞] such that \({p}^{-1} + {q}^{-1} \leq 1\,\), then the inequality
holds which has been first proved by Yu.A. Davydov [146] with some positive constant C ( ≥ 10) . Recently, by improving the approximation technique used in [146], the author and M. Nolde could prove that C = 2 is possible, see also [423] for a different approach. A corresponding estimate with β(σ(ξ), σ(η)) rather than α(σ(ξ), σ(η)) on the right-hand side goes back to K. Yoshihara [522], see also [236] for this and further references.
Let us consider a Voronoi-tessellation \(V (\varPsi ) = \bigcup \nolimits _{i\geq 1}\partial C_{i}(\varPsi )\) generated by a simple stationary PP \(\varPsi =\sum _{i\geq 1}\delta _{X_{i}}\,\), where ∂C i (Ψ) denotes the boundary of the cell C i (Ψ) formed by all point in \({\mathbb{R}}^{d}\) which are closest to the atom X i , i.e. \(C_{i}(\varPsi ) =\{ x \in {\mathbb{R}}^{d} :\| x - X_{i}\| <\| x - X_{j}\|,j\neq i\}\,\), and let denote by \(\mathcal{A}_{\varPsi }(F)\) resp. \(\mathcal{A}_{V (\varPsi )}(F)\) the σ-algebra generated by the PP Ψ restricted to \(F \subset {\mathbb{R}}^{d}\) resp. the σ-algebra generated by the random closed set V (Ψ) ∩ F . With the notation \(F_{a} = {[-a,a]}^{d}\) and Δ = b ∕ 4 the estimate
has been obtained in [230], where R(a, b) is a finite sum of certain void probabilities of the PP Ψ decaying to zero at some rate (depending on a) as b → ∞ . A similar estimate of \(\beta {\Bigl(\mathcal{A}_{\Xi }(F_{a}),\mathcal{A}_{\Xi }(F_{a+b}^{c})\Bigr)}\) could be derived in [236] for stationary grain-germ models \(\Xi = \bigcup \nolimits _{i\geq 1}(\Xi _{i} + X_{i})\,\) in terms of a suitable β-mixing coefficient of the generating stationary PP Ψ ∼ P with intensity λ and the distribution function D(x) = P(diam(Ξ 0) ≤ x) of the diameter of the typical grain Ξ 0 :
Note that \(\beta {\Bigl(\mathcal{A}_{\varPsi }(F_{a+\varDelta }),\mathcal{A}_{\varPsi }(F_{a+3\,\varDelta }^{c})\Bigr)} = 0\) in (4.19) and (4.20) if \(\varPsi \sim \Pi _{\it\lambda \,\nu _{d}}\), i.e. for the Poisson–Voronoi tessellation and for Boolean models. Furthermore, there exist precise estimates of this β-mixing coefficient for Poisson-cluster and Cox processes and some classes of dependently thinned Poisson processes. We only mention that both of the previous estimates can be reformulated with slight modifications in terms of α-mixing coefficients.
In [236] a CLT for geometric functionals of β-mixing random closed sets has been proved. The conditions of this CLT can be expressed more explicitly for germ-grain models due to (4.20). CLT’s for stationary random fields put assumptions on mixing rates derived from mixing coefficients between specific σ-algebras, see [267] in the case of PP’s. An application of α-mixing to study empirical functionals of geostatistically marked point processes can be found in [392]. Besides the frequently used CLT of E. Bolthausen [74] the following CLT (first proved and applied in [230]) presents a meaningful alternative to verify asymptotic normality of estimators in stochastic-geometric models.
Let ξ = { ξ(t), t ∈ V n } be a stationary random field with index set \(V _{n} =\{ t \in {\mathbb{Z}}^{d} : ([0,1{)}^{d} + t) \subset W_{n}\}\), where \(\{W_{n}\,,\,n \in \mathbb{N}\}\) is a convex averaging sequence in \({\mathbb{R}}^{d}\) implying \(\vert V _{n}\vert /\nu _{d}(W_{n})\rightarrow _{n\rightarrow \infty }1\,\). Further, \(\mathcal{A}_{\xi }(F)\) denotes the σ-algebra generated by the random variables \(\{\xi (t),\,t \in F \cap {\mathbb{Z}}^{d}\}\) and \(S_{n} =\sum _{t\in V _{n}}\,\xi (t)\,\).
Theorem 4.9.
Assume that there are two functions \(\beta _{\xi }^{{_\ast}}\) and \(\beta _{\xi }^{{_\ast}{_\ast}}\) on \(\mathbb{N}\) such that
If, for some δ > 0 ,
then the asymptotic variance \({\tau }^{2} =\lim \limits _{n\rightarrow \infty }\,\mathbf{v}\mathbf{a}\mathbf{r}\,S_{n}/\nu _{d}(W_{n}) =\sum _{t\in {\mathbb{Z}}^{d}}\,\mathbf{c}\mathbf{o}\mathbf{v}(\xi (o),\xi (t))\) exists and the normal convergence \({(\nu _{d}(W_{n}))}^{-1/2}\,{\Bigl(\,S_{n} -\vert V _{n}\vert \,\mathbf{E}\xi (o)\,\Bigr)}\mathop\Rightarrow\limits_{n\rightarrow \infty }N(0,{\tau }^{2})\) holds.
Note that the assertion of Theorem 4.9 remains valid if the slightly weaker α-mixing coefficient is used, see [235] and references therein.
On the other hand, there are situations which require the stronger β-mixing coefficient. For example, Ψ ∼ P can be shown to be B k -mixing for any fixed k ≥ 2 if \(\mathbf{E}\varPsi {({[0,1]}^{d})}^{k+\varDelta } < \infty \) and
for some δ > 0 , where the β-mixing coefficient β Ψ : [1, ∞) → [0, 1] is defined as a non-increasing function such that \(\beta _{\varPsi }(r) \geq {\Bigl(\min \{ 1,{ \frac{r} {a}\}\Bigr)}}^{d-1}\,\beta {\Bigl(\mathcal{A}_{ \varPsi }(F_{a}),\mathcal{A}_{\varPsi }(F_{a+r}^{c})\Bigr)}\) for all a, r ≥ 1 . This implies that Ψ ∼ P is Brillinger-mixing if \(\beta _{\varPsi }(r) \leq {e}^{-g(r)}\) with g : [1, ∞) → [0, ∞] satisfying g(r) ∕ log(r)→ r → ∞ ∞ .
4.3.3 Testing CSR Based on Empirical K-Functions
Let \({\Bigl(\widehat{{\it\lambda }^{2}K_{B}}\Bigr)}_{n}(r)\) be any of the empirical K-functions \({\Bigl(\widehat{{\it\lambda }^{2}K_{B}}\Bigr)}_{n,i}(r)\,\), i = 1, 2, 3, introduced and discussed in the above Sect. 4.2.1. Below we formulate two functional CLT’s for the corresponding centered and scaled empirical process on the interval [0, R] when Ψ = { X i , i ≥ 1} is a stationary Poisson process. We distinguish between the cases of known intensity λ and estimated intensity \(\widehat{\it\lambda }_{n}\) which leads to two distinct zero mean Gauss–Markov limit processes (in the sense of weak convergence in the Skorokhod-space D[0, R], see [69]). For both limits the distribution function of the maximal deviation over [0, R] can be calculated. This fact can be used to establish a Kolmogorov–Smirnov-type test for checking the null hypothesis of CSR via testing the suitably scaled maximal deviation of the empirical K-functions from \({\it\lambda }^{2}\,\nu _{d}(B)\,{r}^{d}\) resp. \((\widehat{{\it\lambda }^{2}})_{n}\,\nu _{d}(B)\,{r}^{d}\,\), see (4.17). For the details of the proofs (in the particular case B = B 1(o) ) and some extensions (among them, a Cramér-von Mises-type test for K-functions) the reader is referred to [228].
Theorem 4.10.
Let the stationary Poisson process \(\varPsi \sim \Pi _{\it\lambda \,\nu _{d}}\) with intensity λ > 0 be observed in window \(W_{n} = \times _{i=1}^{d}[0,a_{i}^{(n)}]\) with unboundedly increasing edges. Then
for 0 ≤ r ≤ R , where ⇒ n→∞ stands for weak convergence in the Skorokhod-space D [0,R] . Both weak limits ζ(r) and η(r) for r ∈ [0,R] are Gaussian diffusion processes with zero means and covariance functions
for 0 ≤ s ≤ t ≤ R . In (4.21)\(\,,\,{ d \atop =}\) means stochastic equivalence, W ={ W(t), t ≥ 0} denotes the one-dimensional standard Wiener process and \(L(r) = 2\,\it\lambda \,\nu _{d}(B)\;{r}^{d}/(\,1 + 2\,\it\lambda \,\nu _{d}(B)\;{r}^{d}\,)\) .
Corollary 4.1.
The continuous mapping theorem, see [69], applied to (4.21) implies that
where L = L(R) ( < 1 ) and
and \(\qquad 1 - G(x)\; =\; 4{\Bigl(\,1 - \varPhi (x)\,\Bigr)} + 4\sum \limits _{n=1}^{\infty }{(-1)}^{n}{\Bigl(\,1 - \varPhi (\,(2n + 1)x\,)\,\Bigr)}\,\) .
Remark 4.5.
The relevant quantiles of F L and G are known. Obviously, testing the CSR-property via checking the goodness-of-fit of the K-function seems to be easier when λ is unknown. The convergence of the finite-dimensional distributions of \(\{\zeta _{n}(\cdot ),\,n \in \mathbb{N}\}\) follows from the CLT for m-dependent fields and the tightness in D[0, R] is seen by an exact bound of the mixed fourth-order moment of two consecutive increments. The convergence of the finite-dimensional distributions of \(\{\eta _{n}(\cdot ),\,n \in \mathbb{N}\}\) follows by applying a variant of Stein’s method to an asymptotically degenerate U-statistic, see [228, 231]
In [231] an analogous test of CSR based on the multivariate K-function \(K(r_{1},..,r_{d}) = {2}^{d}\,r_{1} \cdot \ldots \cdot r_{d}\) and its empirical counterpart in case of a Poisson process has been developed. We only sketch the main result in the case of unknown intensity λ. In [231] the case of known λ is also treated in detail.
Let the assumptions of Theorem 4.10 be satisfied. Setting
for \(\mathbf{r} = {(r_{1},\ldots ,r_{d})}^{T} \in [0,\infty {)}^{d}\,\), and
we obtain a sequence \(\{\eta _{n}(\mathbf{r}),\,\mathbf{r} \in {[0,R]}^{d}\}\) of empirical processes belonging to the Skorokhod-space D([0, R]d) of d-parameter càdlàg-processes that converges weakly to a Gaussian random field \(\{\eta (\mathbf{r}),\,\mathbf{r} \in {[0,R]}^{d}\}{ d \atop =} \{\sqrt{{2}^{d+1}}\,W_{d}(\mathbf{r}),\,\mathbf{r} \in {[0,R]}^{d}\,\}\,\), where \(\{\,W_{d}(\mathbf{r}),\,\mathbf{r} \in [0,\infty {)}^{d}\,\}\,\) denotes the d-dimensional standard Wiener sheet with mean value function E W d (r) = 0 and covariance function \(\mathbf{E}W_{d}(\mathbf{s)}W_{d}(\mathbf{t}) = \prod \nolimits _{i=1}^{d}(\,s_{i} \wedge t_{i}\,)\) for \(\mathbf{s} = {(s_{1},\ldots ,s_{d})}^{\top }\,\), \(\mathbf{t} = {(t_{1},\ldots ,t_{d})}^{\top }\,\). Hence, by the continuous mapping theorem it follows that
The α-quantiles of the distribution function
can be determined only approximately via large-scale simulations of the planar Wiener sheet. In this way we found m 0. 95 = 2. 1165, m 0. 99 = 2. 7105, and m 0. 995 = 2. 9313 , where \(G_{2}(m_{\alpha }) = \alpha \,\).
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Heinrich, L. (2013). Asymptotic Methods in Statistics of Random Point Processes. In: Spodarev, E. (eds) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol 2068. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33305-7_4
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