Clustering comparison of point processes with applications to random geometric models

Clustering comparison of point processes with applications to random geometric models

Bartłomiej Błaszczyszyn and Dhandapani Yogeshwaran B. Błaszczyszyn Inria/ENS, 23 av. d’Italie, 75214 Paris France 22email: Bartek.Blaszczyszyn@ens.frD. Yogeshwaran Dept. of Electrical Engineering, Technion – Israel Institute of Technology, Haifa 32000, ISRAEL 44email:

In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the Boolean model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers for Čech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some “sub-Poisson” processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.

1 Introduction

On the one hand, various interesting methods have been developed for studying local and global functionals of geometric structures driven by Poisson or Bernoulli point processes (see MeeRoy96 (); Penrose03 (); Yukich12 ()). On the other hand, as will be shown in the following section, there are many examples of interesting point processes that occur naturally in theory and applications. So, the obvious question arises how much of the theory developed for Poisson or Bernoulli point processes can be carried over to other classes of point processes.

Our approach to this question is based on the comparison of clustering properties of point processes. Roughly speaking, a set of points in clusters if it lacks spatial homogeneity, i.e., one observes points forming groups which are well spaced out. Many interesting properties of random geometric models driven by point processes should depend on the “degree” of clustering. For example, it is natural to expect that concentrating points of a point process in well-spaced-out clusters should negatively impact connectivity of the corresponding random geometric (Gilbert) graph, and that spreading these clustered points “more homogeneously” in the space would result in a smaller critical radius for which the graph percolates. For many other functionals, using similar heuristic arguments one can conjecture whether increase or decrease of clustering will increase or decrease the value of the functional. However, to the best of our knowledge, there has been no systematic approach towards making these arguments rigorous.

The above observations suggest the following program. We aim at identifying a class or classes of point processes, which can be compared in the sense of clustering to a (say homogeneous) Poisson point process, and for which — by this comparison — some results known for the latter process can be extrapolated. In particular, there are point processes which in some sense cluster less (i.e. spread their points more homogeneously in the space) than the Poisson point process. We call them sub-Poisson. Furthermore, we hasten to explain that the usual strong stochastic order (i.e. coupling as a subset of the Poisson process) is in general not an appropriate tool in this context.

Various approaches to mathematical formalisation of clustering will form an important part of this chapter. By formalisation, we mean defining a partial order on the space of point processes such that being smaller with respect to the order indicates less clustering. The most simple approach consists in considering void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. When stronger tools are needed, directionally convex (dcx) ordering of point processes happens to be a good choice, as well as the notion of negative and positive association. Working with these tools, we first give some useful, generic inequalities regarding Laplace transforms of the compared point processes. In the case of dcx-ordering these inequalities can be generalised to dcx functions of shot-noise fields.

Having described the clustering comparison tools, we present several particular results obtained by using them. Then, in more detail, we study percolation in the Boolean model (seen as a random geometric graph) and the SINR graph. In particular, we show how the classical results regarding the existence of a non-trivial phase transition extend to models based on additive functionals of various sub-Poisson point processes. Furthermore, we briefly discuss some applications of the comparison tools to U-statistics of point processes, counts of sub-graphs and simplices in random geometric graphs and simplicial complexes, respectively. We also mention results on Betti numbers of Čech and Vietoris-Rips random complexes generated by sub-Poisson point processes.

Let us conclude the above short motivating introduction by citing an excerpt from a standard reference on stochastic comparison methods by Müller and Stoyan (2002): “It is clear that there are processes of comparable variability. Examples of such processes are a homogeneous Poisson process and a cluster process of equal intensity or two hard-core Gibbs processes of equal intensity and different hard-core distances. It would be fine if these variability differences could be characterized by order relations … [implying], for example, reasonable relationship[s] for second order characteristics such as the pair correlation function.”; cf (Muller02, , page 253). We believe that the results reported in this chapter present one of the first answers to the above call, although “Still much work has to be done in the comparison theory for point processes.” (ibid.)

The present chapter is organised as follows. In Sect. 2 we recall some examples of point processes. The idea is to provide as many as possible examples which admit various clustering comparison orders, described then in Sect. 3. An extensive overview of applications is presented in Sect. 4.

2 Examples of Point Processes

In this section we give some examples of point processes, where our goal is to present them in the context of modelling of clustering phenomena. Note that throughout this chapter, we consider point processes on the -dimensional Euclidean space , , although much of the presented results have straightforward extensions to point processes on an arbitrary Polish space.

2.1 Elementary Models

A first example we probably think of when trying to come up with some spatially homogeneous model is a point process on a deterministic lattice.

Definition 1 (Lattice point process)

By a lattice point process we mean a simple point process whose points are located on the vertices of some deterministic lattice . An important special case is the -dimensional cubic lattice of edge length , where denotes the set of integers. Another specific (two-dimensional) model is the hexagonal lattice on the complex plane given by . The stationary version of a lattice point process can be constructed by randomly shifting the deterministic pattern through a vector uniformly distributed in some fixed cell of the lattice , i.e. Note that the intensity of the stationary lattice point process is equal to the inverse of the cell volume. In particular, the intensity of is equal to , while that of is equal to .

Lattice point processes are usually considered to model “perfect” or “ideal” structures, e.g. the hexagonal lattice on the complex plane is used to study perfect cellular communication networks. We will see however, without further modifications, they escape from the clustering comparison methods presented in Sect. 3.

When the “perfect structure” assumption cannot be retained and one needs a random pattern, then the Poisson point process usually comes as a natural first modelling assumption. We therefore recall the definition of the Poisson process for the convenience of the reader, see also the survey given in baddeley12 ().

Definition 2 (Poisson point process)

Let be a (deterministic) locally finite measure on the Borel sets of . The random counting measure is called a Poisson point process with intensity measure if for every and all bounded, mutually disjoint Borel sets , the random variables are independent, with Poisson distribution , respectively. In the case when has an integral representation , where is some measurable function, we call the intensity field of the Poisson point process. In particular, if is a constant, we call a homogeneous Poisson point process and denote it by .

The Poisson point process is a good model when one does not expect any “interactions” between points. This is related to the complete randomness property of Poisson processes, cf. (DVJI2003, , Theorem 2.2.III). {svgraybox} The homogeneous Poisson point process is commonly considered as a reference model in comparative studies of clustering phenomena.

2.2 Cluster Point Processes — Replicating and Displacing Points

We now present several operations on the points of a point process, which in conjunction with the two elementary models presented above allow us to construct various other interesting examples of point processes. We begin by recalling the following elementary operations.

Superposition of patterns of points consists of set-theoretic addition of these points. Superposition of (two or more) point processes , defined as random counting measures, consists of adding these measures . Superposition of independent point processes is of special interest.

Thinning of a point process consists of suppressing some subset of its points. Independent thinning with retention function defined on , , consists in suppressing points independently, given a realisation of the point process, with probability , which might depend on the location of the point to be suppressed or not.

Displacement consists in, possibly randomised, mapping of points of a point process to the same or another space. Independent displacement with displacement (probability) kernel from to , , consists of independently mapping each point of a given realisation of the original point process to a new, random location in selected according to the kernel .

Remark 1

An interesting property of the class of Poisson point processes is that it is closed with respect to independent displacement, thinning and superposition, i.e. the result of these operations made on Poisson point processes is a Poisson point process, which is not granted in the case of an arbitrary (i.e. not independent) superposition or displacement, see e.g. DVJI2003 (); DVJII2007 ().

Now, we define a more sophisticated operation on point processes that will allow us to construct new classes of point processes with interesting clustering properties.

Definition 3 (Clustering perturbation of a point process)

Let be a point process on and , be two probability kernels from to the set of non-negative integers and , , respectively. Consider the following subset of . Let


where, given ,

  • are independent, non-negative integer-valued random variables with (conditional) distribution ,

  • , are independent vectors of i.i.d. random elements of , with having the conditional distribution .

Note that the inner sum in (1) is interpreted as when .

The random set given in (1) can be considered as a point process on provided it is a locally finite. In what follows, we will assume a stronger condition, namely that the itensity measure of is locally finite (Radon), i.e.


for all bounded Borel sets , where denotes the intensity measure of and


is the mean value of the distribution .


Clustering perturbation of a given parent process consists in independent replication and displacement of the points of , with the number of replications of a given point having distribution and the replicas’ locations having distribution . The replicas of form a cluster.

For obvious reasons, we call a perturbation of  driven by the replication kernel and the displacement kernel . It is easy to see that the independent thinning and displacement operations described above are special cases of clustering perturbation. In what follows we present a few known examples of point processes arising as clustering perturbations of a lattice or Poisson point process. For simplicity we assume that replicas stay in the same state space, i.e. .

Example 1 (Binomial point process)

A (finite) binomial point process has a fixed total number of points, which are independent and identically distributed according to some (probability) measure on . It can be seen as a Poisson point process conditioned to have points, cf. DVJI2003 (). Note that this property might be seen as a clustering perturbation of a one-point process, with deterministic number of point replicas and displacement distribution .

Example 2 (Bernoulli lattice)

The Bernoulli lattice arises as independent thinning of a lattice point process; i.e., each point of the lattice is retained (but not displaced) with some probability and suppressed otherwise.

Example 3 (Voronoi-perturbed lattices)

These are perturbed lattices with displacement kernel , where the distribution is supported on the Voronoi cell of vertex of the original (unperturbed) lattice . In other words, each replica of a given lattice point gets independently translated to some random location chosen in the Voronoi cell of the original lattice point. Note that one can also choose other bijective, lattice-translation invariant mappings of associating lattice cells to lattice points; e.g. associate a given cell of the square lattice on the plane to its “south-west” corner.

By a simple perturbed lattice we mean the Voronoi-perturbed lattice whose points are uniformly translated in the corresponding cells, without being replicated. Interestingly enough, the Poisson point process with some intensity measure can be constructed as a Voronoi-perturbed lattice. Indeed, it is enough to take the Poisson replication kernel given by and the displacement kernel with ; cf. Exercise 1. Keeping the above displacement kernel and replacing the Poisson distribution in the replication kernel by some other distributions convexly smaller or larger than the Poisson distribution, one gets the following two particular classes of Voronoi-perturbed lattices, clustering their points less or more than the Poisson point process (in a sense that will be formalised in Sect. 3).

Sub-Poisson Voronoi-perturbed lattices are Voronoi-perturbed lattices such that is convexly smaller than . Examples of distributions convexly smaller than are the hyper-geometric distributions , , and the binomial distributions. , , which can be ordered as follows:


for ; cf. Whitt1985 (). Recall that has the probability mass function (), whereas has the probability mass function (); cf. Exercise 2.

Super-Poisson Voronoi-perturbed lattices are Voronoi-perturbed lattices with convexly larger than . Examples of distributions convexly larger than are the negative binomial distribution with and the geometric distribution distribution , which can be ordered in the following way:


with , , and , where the largest distribution in (5) is a mixture of geometric distributions having mean . Note that any mixture of Poisson distributions having mean is in cx-order larger than . Furthermore, recall that the probability mass functions of and are given by and , respectively.

Example 4 (Generalised shot-noise Cox point processes)

These are clustering perturbations of an arbitrary parent point process , with replication kernel , where is the Poisson distribution and is the mean value given in (3). Note that in this case, given , the clusters (i.e. replicas of the given parent point) form independent Poisson point process , . This special class of Cox point processes (cf. Sect. 2.3) has been introduced in Moller05 ().

Example 5 (Poisson-Poisson cluster point processes)

This is a special case of the generalised shot-noise Cox point processes, with the parent point process being Poisson, i.e. for some intensity measure . A further special case is often discussed in the literature, where the displacement kernel is such that is the uniform distribution in the ball of some given radius . It is called the Matérn cluster point process. If is symmetric Gaussian, then the resulting Poisson-Poisson cluster point process is called a (modified) Thomas point process.

Example 6 (Neyman-Scott point process)

These point processes arise as a clustering perturbation of a Poisson parent point process , with arbitrary (not necessarily Poisson) replication kernel .

2.3 Cox Point Processes— Mixing Poisson Distributions

We now consider a rich class of point processes known also as doubly stochastic Poisson point process, which are often used to model patterns exhibiting more clustering than the Poisson point process.

Definition 4 (Cox point process)

Let be a random locally finite (non-null) measure on . A Cox point process on generated by is defined as point process having the property that conditioned on is the Poisson point process . Note that is called the random intensity measure of . In case when the random measure has an integral representation , with being a random field, we call this field the random intensity field of the Cox process. In the special case that for all , where is a (non-negative) random variable and a (non-negative) deterministic function, the corresponding Cox point process is called a mixed Poisson point process.


Cox processes may be seen as a result of an operation transforming some random measure into a point process , being a mixture of Poisson processes.

In Sect. 2.2, we have already seen that clustering perturbation of an arbitrary point process with Poisson replication kernel gives rise to Cox processes (cf. Example 4), where Poisson-Poisson cluster point processes are special cases with Poisson parent point process. This latter class of point processes can be naturally extended by replacing the Poisson parent process by a Lévy basis.

Definition 5 (Lévy basis)

A collection of real-valued random variables , where denotes the family of bounded Borel sets in , is said to be a Lévy basis if the are infinitely divisible random variables and for any sequence , , of disjoint bounded Borel sets in , are independent random variables (complete independence property), with almost surely provided that is bounded.

In this chapter, we shall consider only non-negative Lévy bases. We immediately see that the Poisson point process is a special case of a Lévy basis. Many other concrete examples of Lévy bases can be obtained by “attaching” independent, infinitely divisible random variables to a deterministic, locally finite sequence of (fixed) points in and letting . In particular, clustering perturbations of a lattice, with infinitely divisible replication kernel and no displacement (i.e. , where is the Dirac measure at ) are Lévy bases. Recall that any degenerate (deterministic), Poisson, negative binomial, gamma as well as Gaussian, Cauchy, Student’s distribution are examples of infinitely divisible distributions.

It is possible to define an integral of a measurable function with respect to a Lévy basis (even if the latter is not always a random measure; see Hellmund08 () for details) and consequently consider the following classes of Cox point processes.

Example 7 (Lévy-based Cox point process)

Consider a Cox point process with random intensity field that is an integral shot-noise field of a Lévy basis, i.e. , where is a Lévy basis and is some non-negative function almost surely integrable with respect to .

Example 8 (Log-Lévy-based Cox point process)

These are Cox point processes with random intensity field given by , where and satisfy the same conditions as above.

Both Lévy- and log-Lévy-based Cox point processes have been introduced in Hellmund08 (), where one can find many examples of these processes. We still mention another class of Cox point processes considered in Moller98 ().

Example 9 (Log-Gaussian Cox point process)

Consider a Cox point process whose random intensity field is given by where is a Gaussian random field.

2.4 Gibbs and Hard-Core Point Processes

Gibbs and hard-core point processes are two further classes of point processes, which should appear in the context of modelling of clustering phenomena.

Roughly speaking Gibbs point processes are point processes having a density with respect to the Poisson point process. In other words, we obtain a Gibbs point process, when we “filter” Poisson patterns of points, giving more chance to appear for some configurations and less chance (or completely suppressing) some others. A very simple example is a Poisson point process conditioned to obey some constraint regarding its points in some bounded Borel set (e.g. to have some given number of points there). Depending on the “filtering” condition we may naturally create point processes which cluster more or less than the Poisson point process.

Hard-core point processes are point process in which the points are separated from each other by some minimal distance, hence in some sense clustering is “forbidden by definition”.

However, we will not give precise definitions, nor present particular examples from these classes of point processes, because, unfortunately, we do not have yet interesting enough comparison results for them, to be presented in the remaining part of this chapter.

2.5 Determinantal and Permanental Point Process

We briefly recall two special classes of point processes arising in random matrix theory, combinatorics, and physics. They are “known” to cluster their points, less or more, respectively, than the Poisson point process.

Definition 6 (Determinantal point process)

A simple point process on is said to be a determinantal point process with a kernel function with respect to a Radon measure on if the joint intensities of the factorial moment measures of the point process with respect to the product measure satisfy for all , where stands for a matrix with entries and denotes the determinant of the matrix.

Definition 7 (Permanental point process)

Similar to the notion of a determinantal point process, one says that a simple point process is a permanental point process with a kernel function with respect to a Radon measure on if the joint intensities of the point process with respect to satisfy for all , where stands for the permanent of a matrix. From (Ben06, , Proposition 35 and Remark 36), we know that each permanental point process is a Cox point process.

Naturally, the kernel function needs to satisfy some additional assumptions for the existence of the point processes defined above. We refer to (Ben09, , Chap. 4) for a general framework which allows to study determinantal and permanental point processes, see also Ben06 (). Regarding statistical aspects and simulation methods for determinantal point processes, see Lavancier12 ().

Here is an important example of a determinantal point process recently studied on the theoretical ground (cf. e.g. Goldman2010 ()) and considered in modelling applications (cf. Miyoshi2012 ()).

Example 10 (Ginibre point process)

This is the determinantal point process on with kernel function , , , with respect to the measure .

Exercise 1

Let be a simple point process on . Consider its cluster perturbation defined in (1) with the Poisson replication kernel , where is the Voronoi cell of in , and the displacement kernel , for some given deterministic Radon measure on . Show that is Poisson with intensity measure .

Exercise 2

Prove (4) and (5) by showing the logarithmic concavity of the ratio of the respective probability mass functions, which implies increasing convex order and, consequently, cx-order provided the distributions have the same means.

3 Clustering Comparison Methods

Let us begin with the following informal definitions. {svgraybox} A set of points is spatially homogeneous if approximately the same numbers of points occur in any spherical region of a given volume. A set of points clusters if it lacks spatial homogeneity; more precisely, if one observes points arranged in groups being well spaced out.

Looking at Fig. 1, it is intuitively obvious that (realisations of) some point processes cluster less than others. However, the mathematical formalisation of such a statement appears not so easy. In what follows, we present a few possible approaches. We begin with the usual statistical descriptors of spatial homogeneity, then show how void probabilities and moment measures come into the picture, in particular in relation to another notion useful in this context: positive and negative association. Finally we deal with directionally convex ordering of point processes.

This kind of organisation roughly corresponds to presenting ordering methods from weaker to stronger ones; cf. Fig. 2. We also show how the different examples presented in Sect. 2 admit these comparison methods, mentioning the respective results in their strongest versions. We recapitulate results regarding comparison to the Poisson process in Fig. 3.

simple perturbed lattice Poisson point process Cox point process
Fig. 1: From left to right : patterns sampled from a simple perturbed lattice (cf. Example 3), Poisson point process and a doubly stochastic Poisson (Cox) point process, all with the same mean number of points per unit area.

3.1 Second-order statistics

In this section we restrict ourselves to the stationary setting.

Ripley’s K-Function

One of the most popular functions for the statistical analysis of spatial homogeneity is Ripley’s K-function defined for stationary point processes (cf. stoyetal95 ()). Assume that is a stationary point process on with finite intensity . Then

where denotes the Lebesgue measure of a bounded Borel set , assuming that . Due to stationarity, the definition does not depend on the choice of .

The value of can be interpreted as the average number of “extra” points observed within the distance from a randomly chosen (so-called typical) point. Campbell’s formula from Palm theory of stationary point processes gives a precise meaning to this statement. Consequently, for a given intensity , the more one finds points of a point process located in clusters of radius , the larger the value of is, whence a first clustering comparison method follows. {svgraybox} Larger values of Ripley’s K-function indicate more clustering “at the cluster-radius scale” . For the (homogeneous) Poisson process on , which is often considered as a “reference model” for clustering, we have , where is the volume of the unit ball in . Note here that describes clustering characteristics of the point process at the (cluster radius) scale . Many point processes, which we tend to think that they cluster less or more than the Poisson point process, in fact are not comparable in the sense of Ripley’s K-function (with the given inequality holding for all , neither, in consequence, in any stronger sense considered later in this section), as we can see in the following simple example.

The following result of D. Stoyan from 1983 can be considered as a precursor to our theory of clustering comparison. It says that the convex ordering of Ripley’s K-functions implies ordering of variances of number of observed points. We shall see in Remark 2 that variance bounds give us simple concentration inequalities for the distribution of the number of observed points. These inequalities help to control clustering. We will develop this idea further in Section 3.2 and 3.3 showing that using moment measures and void probabilities one can obtain stronger, exponential concentration equalities.

Proposition 1 ((Stoyan1983inequalities, , Corollary 1))

Consider two stationary, isotropic point processes and of the same intensity, with the Ripley’s functions and , respectively. If i.e., for all decreasing convex then for all compact, convex .

Exercise 3

For the stationary square lattice point process on the plane with intensity (cf. Definition 1), compare and for .

From Exercise 3, one should be able to see that though the square lattice is presumably more homogeneous (less clustering) than the Poisson point process of the same intensity, the differences of the values of their K-functions alternate between strictly positive and strictly negative. However, we shall see later that (cf. Example 18) this will not be the case for some perturbed lattices, including the simple perturbed ones and thus they cluster less than the Poisson point process in the sense of Ripley’s K-function (and even in a much stronger sense). We will also discuss point processes clustering more than the Poisson processes in this sense.

Pair Correlation Function

Another useful characteristic for measuring clustering effects in stationary point processes is the pair correlation function . It is related to the probability of finding a point at a given distance from another point and can be defined as

where is the intensity of the point process and is its joint second-order intensity; i.e. the density (if it exists, with respect to the Lebesgue measure) of the second-order factorial moment measure (cf. Sect. 3.2).


Similarly as for Ripley’s K-function, we can say that larger values of the pair correlation function indicate more clustering “around” the vector .

For stationary point processes the following relation holds between functions  and 

which simplifies to

in the case of isotropic processes; cf (stoyetal95, , Eq. (4.97), (4.98)).

For a Poisson point process , we have that . Again, it is not immediate to find examples of point processes whose pair correlation functions are ordered for all values of . Examples of such point processes will be provided in the following sections.

Exercise 4

Show that ordering of pair correlation functions implies ordering of Ripley’s K-functions, i.e., for two stationary point processes with for almost all , it holds that for all .

Though Ripley’s K-function and the pair-correlation function are very simple to compute, they define only a pre-odering of point processes, because their equality does not imply equality of the underlying point processes. We shall now present some possible definitions of partial ordering of point processes that capture clustering phenomena.

3.2 Moment Measures

Recall that the measure defined by

for all (not necessarily disjoint) bounded Borel sets () is called the -th order moment measure of . For simple point processes, the truncation of the measure to the subset is equal to the -th order factorial moment measure . Note that expresses the expected number of -tuples of points of the point process in a given set .


In the class of point processes with some given intensity measure , larger values and of the (factorial) moment measures and , respectively, indicate point processes clustering more in . A first argument we can give to support the above statement is considered in Exercise 5 below.

Exercise 5

Show that comparability of for all bounded Borel sets implies a corresponding inequality for the pair correlation functions and hence Ripley’s K-functions.

Remark 2

For a stronger justification of the relationship between moment measures and clustering, we can use concentration inequalities, which give upper bounds on the probability that the random counting measure deviates from its intensity measure .

Smaller deviations can be interpreted as supportive for spatial homogeneity. To be more specific, using Chebyshev’s inequality we have

for all bounded Borel sets , and . Thus, for point processes of the same mean measure, the second moments or the Ripley’s functions (via Proposition 1) allow to compare their clustering. Similarly, using Chernoff’s bound, we get that


for any . Both concentration inequalities give smaller upper bounds for the probability of the deviation from the mean (the upper deviation in the case of Chernoff’s bound) for point processes clustering less in the sense of higher-order moment measures. We will come back to this idea in Propositions 2 and 4 below.

In Sect. 4 we will present results, in particular regarding percolation properties of point processes, for which it its enough to be able to compare factorial moment measures of point processes. We shall note casually that restricted to a ”nice” class of point processes, the factorial moment measures uniquely determine the point process and hence the ordering defined via comparison of factorial moment measures is actually a partial order on this nice class of point processes.

We now concentrate on comparison to the Poisson point process. Recall that for a general Poisson point process we have for all , where is the intensity measure . In this regard, we define the following class of point processes clustering less (or more) than the Poisson point process with the same intensity measure.

Definition 8 (-weakly sub-Poisson point process)

A point process is said to be weakly sub-Poisson in the sense of moment measures (-weakly sub-Poisson for short) if


for all and all mutually disjoint bounded Borel sets . When the reversed inequality in (7) holds, we say that is weakly super-Poisson in the sense of moment measures (-weakly super-Poisson for short).

In other words, -weakly sub-Poisson point processes have factorial moment measures smaller than those of the Poisson point process with the same intensity measure. Similarly, -weakly super-Poisson point processes have factorial moment measures larger than those of the Poisson point process with the same intensity measure. We also remark that the notion of sub- and super-Poisson distributions is used e.g. in quantum physics and denotes distributions for which the variance is smaller (respectively larger) than the mean. Our notion of -weak sub- and super-Poissonianity is consistent with (and stronger than) this definition. In quantum optics, e.g. sub-Poisson patterns of photons appear in resonance fluorescence, where laser light gives Poisson statistics of photons, while the thermal light gives super-Poisson patterns of photons; cf. Photon_antibunchin_wikipedia ().

Exercise 6

Show that -weakly sub- (super-) Poisson point processes have moment measures smaller (larger) than those of the corresponding Poisson point process. Hint. Recall that the moment measures of a general point process can be expressed as non-negative combinations of products of its (lower-dimensional) factorial moment measures (cf. DVJI2003 () Exercise 5.4.5, p. 143).

Here is an easy, but important consequence of the latter observation regarding Laplace transforms “in the negative domain”, i.e. functionals , where

for non-negative functions on , which include as a special case the functional appearing in the “upper” concentration inequality (6). By Taylor expansion of the exponential function at and the well-known expression of the Laplace functional of the Poisson point process with intensity measure which can be recognised in the right-hand side of (8), the following result is obtained.

Proposition 2

Assume that is a simple point process with locally bounded intensity measure and consider . If is -weakly sub-Poisson, then


If is -weakly super-Poisson, then the reversed inequality is true.

The notion of weak sub(super)-Poissonianity is closely related to negative and positive association of point processes, as we shall see in Sect. 3.4 below.

3.3 Void Probabilities

The celebrated Rényi theorem says that the void probabilities of point processes, evaluated for all bounded Borel Sets characterise the distribution of a simple point process. They also allow an easy comparison of clustering properties of point processes by the following interpretation: a point process having smaller void probabilities has less chance to create a particular hole (absence of points in a given region). {svgraybox} Larger void probabilities indicate point processes with stronger clustering. Using void probabilities in the study of clustering is complementary to the comparison of moments.

Remark 3

An easy way to see the complementarity of voids and moment measures consists in using again Chernoff’s bound to obtain the following “lower” concentration inequality (cf. Remark 2)


which holds for any , and noting that

is the void probability of the point process obtained from by independent thinning with retention probability . It is not difficult to show that ordering of void probabilities of simple point processes is preserved by independent thinning (cf. dcx-perc ()) and thus the bound in (9) is smaller for point processes less clustering in the latter sense. We will come back to this idea in Propositions 3 and 4. Finally, note that and thus, in conjunction with what was said above, comparison of void probabilities is equivalent to the comparison of one-dimensional Laplace transforms of point processes for non-negative arguments.

In Sect. 4, we will present results, in particular regarding percolation properties, for which it is enough to be able to compare void probabilities of point processes. Again, because of Rényi’s theorem, we have that ordering defined by void probabilities is a partial order on the space of simple point processes.

-Weakly Sub(Super)-Poisson Point Processes

Recall that a Poisson point process can be characterised as having void probabilities of the form , with being the intensity measure of . In this regard, we define the following classes of point processes clustering less (or more) than the Poisson point process with the same intensity measure.

Definition 9 (-weakly sub(super)-Poisson point process)

A point process is said to be weakly sub-Poisson in the sense of void probabilities (-weakly sub-Poisson for short) if


for all Borel sets . When the reversed inequality in (10) holds, we say that is weakly super-Poisson in the sense of void probabilities (-weakly super-Poisson for short).

In other words, -weakly sub-Poisson point processes have void probabilities smaller than those of the Poisson point process with the same intensity measure. Similarly, -weakly super-Poisson point processes have void probabilities larger than those of the Poisson point process with the same intensity measure.

Example 11

It is easy to see by Jensen’s inequality that all Cox point processes are -weakly super-Poisson.

By using a coupling argument as in Remark 3, we can derive an analogous result as in Proposition 2 for -weakly sub-Poisson point processes.

Proposition 3dcx-perc ())

Assume that is a simple point process with locally bounded intensity measure . Then is -weakly sub-Poisson if and only if (8) holds for all functions .

Combining Void Probabilities and Moment Measures

We have already explained why the comparison of void probabilities and moment measures are in some sense complementary. Thus, it is natural to combine them, whence the following definition is obtained.

Definition 10 (Weakly sub- and super-Poisson point process)

We say that is weakly sub-Poisson if is -weakly sub-Poisson and -weakly sub-Poisson. Weakly super-Poisson point processes are defined in the same way.

The following remark is immediately obtained from Propositions 2 and 3.

Remark 4

Assume that is a simple point process with locally bounded intensity measure . If is weakly sub-Poisson then (8) holds for any of constant sign ( or ). If is weakly super-Poisson, then in (8) the reversed inequality holds for such .

Example 12

It has been shown in dcx-clust () that determinantal and permanental point process (with trace-class integral kernels) are weakly sub-Poisson and weakly super-Poisson, respectively.

Other examples (admitting even stronger comparison properties) will be given in Sect. 3.4 and 3.5.

As mentioned earlier, using the ordering of Laplace functionals of weakly sub-Poisson point processes, we can extend the concentration inequality for Poisson point processes to this class of point processes. In the discrete setting, a similar result is proved for negatively associated random variables in Dubhashi96 (). A more general concentration inequality for Lipschitz functions is known in the case of determinantal point processes (Pemantle11 ()).

Proposition 4

Let be a simple stationary point process with unit intensity which is weakly sub-Poisson, and let be a Borel set of Lebesgue measure . Then, for any there exists an integer such that for

Exercise 7

Prove Proposition 4. Hint. Use Markov’s inequality, Propositions 2 and  3 along with the bounds for the Poisson case known from (Penrose03, , Lemmas 1.2 and 1.4).

Note that the bounds we have suggested to use are the ones corresponding to the Poisson point process. For specific weakly sub-Poisson point processes, one expects an improvement on these bounds.

3.4 Positive and Negative Association

Denote covariance of (real-valued) random variables by .

Definition 11 ((Positive) association of point processes)

A point process is called associated if


for any finite collection of bounded Borel sets and (componentwise) increasing functions; cf. BurtonWaymire1985 ().

The property considered in (11) is also called positive association, or the FKG property. The theory for the opposite property is more tricky, cf. pemantle00 (), but one can define it as follows.

Definition 12 (Negative association)

A point process is called negatively associated if

for any finite collection of bounded Borel sets such that and increasing functions.

Both definitions can be straightforwardly extended to arbitrary random measures, where one additionally assumes that are continuous and increasing functions. Note that the notion of association or negative association of point processes does not induce any ordering on the space of point processes. Though, association or negative association have not been studied from the point of view of stochastic ordering, it has been widely used to prove central limit theorems for random fields (see Bulinski_Spodarev2012central ()).


Positive and negative association can be seen as clustering comparison to Poisson point process.

The following result, supporting the above statement, has been proved in dcx-clust (). It will be strengthened in the next section (see Proposition 12)

Proposition 5

A negatively associated, simple point process with locally bounded intensity measure is weakly sub-Poisson. A (positively) associated point process with a diffuse locally bounded intensity measure is weakly super-Poisson.

Exercise 8

Prove that a (positively) associated point process with a diffuse locally bounded intensity measure is -weakly super-Poisson. Show a similar statement for negatively associated point processes as well.

Example 13

From (BurtonWaymire1985, , Th. 5.2), we know that any Poisson cluster point process is associated. This is a generalisation of the perturbation approach of a Poisson point process considered in (1) having the form with being arbitrary i.i.d. (cluster) point processes. In particular, the Neyman-Scott point process (cf. Example 6) is associated. Other examples of associated point processes given in BurtonWaymire1985 () are Cox point processes with random intensity measures being associated.

Example 14

Determinantal point processes are negatively associated (see (Ghosh12a, , cf. Corollary 6.3)).

We also remark that there are negatively associated point processes, which are not weakly sub-Poisson. A counterexample given in dcx-clust () (which is not a simple point process, showing that this latter assumption cannot be relaxed in Proposition 5) exploits (JoagDev1983, , Theorem 2), which says that a random vector having a permutation distribution (taking as values all permutations of a given deterministic vector with equal probabilities) is negatively associated.

Exercise 9

Show that the binomial point process (cf. Example 1) and the simple perturbed lattice (cf. Example 3) are negatively associated.

3.5 Directionally Convex Ordering

Definitions and Basic Results

In this section, we present some basic results on directionally convex ordering of point processes that will allow us to see this order also as a tool to compare clustering of point processes.

A Borel-measurable function is said to be directionally convex (dcx) if for any , we have that , where is the discrete differential operator, with denoting the canonical basis vectors of . In the following, we abbreviate increasing and dcx by idcx and decreasing and dcx by ddcx (see (Muller02, , Chap. 3)). For random vectors and of the same dimension, is said to be smaller than in dcx order (denoted ) if for all being dcx such that both expectations in the latter inequality are finite. Real-valued random fields are said to be dcx ordered if all finite-dimensional marginals are dcx ordered.

Definition 13 (dcx-order of point processes)

Two point processes and are said to be dcx-ordered, i.e. , if for any and bounded Borel sets in , it holds that .

The definition of comparability of point processes is similar for other orders, i.e. those defined by functions. It is enough to verify the above conditions for mutually disjoint, cf. snorder (). In order to avoid technical difficulties, we will consider only point processes whose intensity measures are locally finite. For such point processes, the dcx-order is a partial order.

Remark 5

It is easy to see that implies the equality of their intensity measures, i.e: for any bounded Borel set as both and are dcx functions.

We argue that, dcx-ordering is also useful in clustering comparison of point processes. {svgraybox} Point processes larger in dcx-order cluster more, whereas point processes larger in idcx-order cluster more while having on average more points, and point processes larger in ddcx-order cluster more while having on average less points.

The two statements of the following result were proved in snorder () and dcx-clust (), respectively. They show that dcx-ordering is stronger than comparison of moments measures and void probabilities considered in the two previous sections.

Proposition 6

Let and be two point process on . Denote their moment measures by () and their void probabilities by , , respectively.

  1. If then for all bounded Borel sets , provided that is -finite for , .

  2. If then for all bounded Borel sets .

Exercise 10

Show that is a dcx-function and is a convex function. Using these facts to prove the above proposition.

Note that the -finiteness condition considered in the first statement of Proposition 6 is missing in snorder (); see (Yogesh_thesis, , Proposition 4.2.4) for the correction. An important observation is that the operation of clustering perturbation introduced in Sect. 2.2 is dcx monotone with respect to the replication kernel in the following sense; cf. dcx-clust ().

Proposition 7

Consider a point process with locally finite intensity measure and its two perturbations () satisfying condition (2), and having the same displacement kernel and possibly different replication kernels , , respectively. If (which means convex ordering of the conditional distributions of the number of replicas) for -almost all then .

Thus clustering perturbations of a given point process provide many examples of point process comparable in dcx-order. Examples of convexly ordered replication kernels have been given in Example 3.

Another observation, proved in snorder (), says that the operations transforming some random measure into a Cox point process (cf. Definition 4) preserves the dcx-order.

Proposition 8

Consider two random measures and on . If then .

The above result, combined with further results on comparison of shot-noise fields presented in Sect. 3.5 will allow us to compare many Cox point processes (cf. Example 15).

Comparison of Shot-Noise Fields

Many interesting quantities in stochastic geometry can be expressed by additive or extremal shot-noise fields. They are also used to construct more sophisticated point process models. For this reason, we state some results on dcx-ordering of shot-noise fields that are widely used in applications.

Definition 14 (Shot-noise fields)

Let be any (non-empty) index set. Given a point process on and a response function which is measurable in the first variable, then the (integral) shot-noise field is defined as


and the extremal shot-noise field is defined as


As we shall see in Sect. 4.2 (and also in the proof of Proposition 11) it is not merely a formal generalisation to take being an arbitrary set. Since the composition of a dcx-function with an increasing linear function is still dcx, linear combinations of for finitely many bounded Borel sets (i.e. for ) preserve the dcx-order. An integral shot-noise field can be approximated by finite linear combinations of ’s and hence justifying continuity, one expects that integral shot-noise fields preserve dcx-order as well. This type of important results on dcx-ordering of point processes is stated below.

Proposition 9

(snorder, , Theorem 2.1)) Let and be arbitrary point processes on . Then, the following statements are true.

  1. If , then .

  2. If , then , provided that , for all , .

The results of Proposition 9, combined with those of Proposition 8 allow the comparison of many Cox processes.

Example 15 (Comparable Cox point processes)

Let and be two Lévy-bases with mean measures and , respectively. Note that (). This can be easily proved using complete independence of Lévy bases and Jensen’s inequality. In a sense, the mean measure “spreads” (in the sense of dcx) the mass better than the corresponding completely independent random measure . Furthermore, consider the random fields and on given by , for some non-negative kernel , and assume that these fields are a.s. locally Riemann integrable. Denote by and the corresponding Lévy-based and log-Lévy-based Cox point process. The following inequalities hold.

  1. If , then provided that, in case of dcx, for all bounded Borel sets .

  2. If , then .

Suppose that are two Gaussian random fields on and denote by , () the corresponding log-Lévy-based Cox point processes. Then the following is true.

  1. If (as random fields), then .

Note that the condition in the third statement is equivalent to for all and for all . An example of a parametric dcx-ordered family of Gaussian random fields is given in miyoshi04 ().

Let , be two point processes on and denote by , the generalised shot-noise Cox point processes (cf. Example 4) being clustering perturbations of , respectively, with the same (Poisson) replication kernel and with displacement distributions having density for all . Then, the following result is true.

  1. If , then provided that, in case of dcx, for all , where is the (common) intensity measure of and .

Proposition 9 allows us to compare extremal shot-noise fields using the following well-known representation where is an additive shot-noise field with response function taking values in Noting that is a dcx-function, we get the following result.

Proposition 10(snorder, , Proposition 4.1))

Let . Then for any and for all , it holds that

An example of application of the above result is the comparison of capacity functionals of Boolean models whose definition we recall first.

Definition 15 (Boolean model)

Given (the distribution of) a random closed set and a point process , a Boolean model with the point process of germs and the typical grain , is given by the random set , where , , and is a sequence of i.i.d. random closed sets distributed as . We call a fixed grain if there exists a (deterministic) closed set such that a.s. In the case of spherical grains, i.e. , where is the origin of and a constant, we denote the corresponding Boolean model by .

A commonly made technical assumption about the distributions of and is that for any compact set , the expected number of germs such that is finite. This assumption, called “local finiteness of the Boolean model” guarantees in particular that is a random closed set in . The Boolean models considered throughout this chapter will be assumed to have the local finiteness property.

Proposition 11 ((perc-dcx, , Propostion 3.4))

Let , be two Boolean models with point processes of germs , , respectively, and common distribution of the typical grain . Assume that and