Properties of Poisson point processes

Laplace functional

Let \sX = \R^d be the d-dimensional Euclidean space. Recall that all the random points are unique for a simple point process S:\Omega\to\sX^\N , and hence S can also be considered as a set of countable points in \sX. Let N:\Omega\to\Z_+^{\sB(\sX)} be the counting process associated with the simple point process S.

Remark 1. We observe that dN(x) = 0 for all x \notin S and dN(x) = \delta_x\SetIn{x \in S}. Hence, for any Borel measurable function f : \sX \to \R and bounded A \in \sB(\sX), we have \int_{x \in A}f(x)dN(x) = \sum_{x \in S\cap A}f(x). % = \sum_{S_i \in A}f(S_i).

Definition 1. The Laplace functional \sL_S: \R_+^\sX\to\R_+ of a point process S:\Omega\to\sX^\N and associated counting process N:\Omega\to\Z_+^{\sB(\sX)} is defined for all non-negative Borel measurable function f: \sX \to \R_+ as

Remark 2. For a simple function f = \sum_{i = 1}^{k}t_i\Ind{A_i}, we can write the Laplace functional as a function of the vector (t_1, t_2, \dots, t_k), \sL_{S}(f) = \E\exp\left(-\sum_{i=1}^kt_i\int_{A_i}dN(x)\right) = \E\exp\left(-\sum_{i=1}^kt_iN(A_i)\right). We observe that this is a joint Laplace transform of the random vector (N(A_1), \dots, N(A_k)). This way, one can compute all finite dimensional distribution of the counting process N.

Proposition 2. The Laplace functional of a Poisson point process S:\Omega\to\sX^\N with intensity measure \Lambda:\sB(\sX)\to\R_+ evaluated at any non-negative Borel measurable function f:\sX\to\R_+, is \begin{equation*} \sL_{S}(f) = \exp\left(-\int_{\sX}(1-e^{-f(x)})d\Lambda(x)\right). \end{equation*}

Proof. Proof. For a bounded Borel measurable set A \in \sB(\sX), consider the truncated function g = f\Ind{A}. Then, \begin{equation*} \sL_{S}(g) = \E\exp(-\int_{\sX}g(x)dN(x)) = \E\exp(-\int_{A}f(x)dN(x)). \end{equation*} Clearly dN(x) = \delta_{x}\SetIn{x \in S} and hence we can write \sL_{S}(g) = \E\exp\left(-\sum_{x \in S\cap A}f(x)\right). We know that the probability of N(A) = |S \cap A| = n points in set A is given by Given there are n points in set A, the density of n point locations are independent and given by Hence, we can write the Laplace functional as Result follows from taking increasing sequences of sets A_k \uparrow \sX and monotone convergence theorem. ◻

Superposition of point processes

Definition 3. Let S^k:\Omega\to\sX^\N be a simple point process with intensity measures \Lambda_k:\sB(\sX)\to\R_+ and counting process N_k:\Omega\to\Z_+^{\sB(\sX)}, for each k \in \N. The superposition of point processes (S^k: k \in \N) is defined as a point process S \triangleq \cup_kS^k.

Remark 3. The counting process associated with superposition point process S: \Omega\to\sX^\N is given by N: \Omega\to\Z_+^{\sB(\sX)} defined by N \triangleq \sum_kN_k, and the intensity measure of point process S is given by \Lambda: \sB(\sX)\to\R_+ defined by \Lambda = \sum_k\Lambda_k from monotone convergence theorem.

Remark 4. The superposition process S is simple iff \sum_kN_k is locally finite.

Theorem 4. The superposition of independent Poisson point processes (S^k: k \in \N) with intensities (\Lambda_k: k \in \N) is a Poisson point process with intensity measure \sum_k \Lambda_k if and only if the latter is a locally finite measure.

Proof. Proof. Consider the superposition S = \cup_kS^k of independent Poisson point processes S^k \in \sX with intensity measures \Lambda_k. We will prove just the sufficiency part this theorem. We assume that \sum_k\Lambda_k is locally finite measure. It is clear that N(A) = \sum_kN_k(A) is finite by locally finite assumption, for all bounded sets A \in \sB(\sX). In particular, we have dN(x) = \sum_kdN_k(x) for all x \in \sX. From the monotone convergence theorem and the independence of counting processes, we have for a non-negative Borel measurable function f: \sX \to \R_+, ◻

Thinning of point processes

Definition 5. Consider a probability retention function p: \sX \to [0,1] and an independent Bernoulli point retention process Y:\Omega \to \set{0,1}^\sX such that \E Y(x) = p(x) for all x \in \sX. The thinning of point process S: \Omega \to \sX^\N with the probability retention function p: \sX \to [0,1] is a point process S^{(p)}: \Omega \to \sX^\N defined by where Y(S_n) is an independent indicator for the retention of each point S_n and \E[Y(S_n)\given S_n] = p(S_n).

Theorem 6. The thinning of a Poisson point process S:\Omega\to\sX^\N of intensity measure \Lambda:\sB(\sX)\to\R_+ with the retention probability function p:\sX\to[0,1], yields a Poisson point process S^{(p)}:\Omega\to\sX^\N of intensity measure \Lambda^{(p)}:\sB(\sX)\to\R_+ defined for all bounded A \in \sB(\sX) as \Lambda^{(p)}(A) \triangleq \int_A p(x) d\Lambda(x).

Proof. Proof. Let A \in \sB(\sX) be a bounded Boreal measurable set, and let f: \sX \to \R_+ be a non-negative function. Let N^{(p)} be the associated counting process to the thinned point process S^{(p)}. Hence, for any bounded set A \in \sB(\sX), we have N^{(p)}(A) = \sum_{x \in S \cap A}Y(x). That is, dN^{(p)}(x) = \delta_{x}Y(x)\SetIn{x \in S}. Therefore, for any non-negative function g(x) = f(x)\SetIn{x \in A}, we can write \int_{x \in \sX}g(x)dN^{(p)}(x) = \int_{x \in A}f(x)dN^{(p)}(x) = \sum_{x \in S\cap A}f(x)Y(x). We can write the Laplace functional of the thinned point process S^{(p)} for the non-negative function g(x) = f(x)\SetIn{x \in A}, as The first equality follows from the definition of Laplace functional and taking nested expectations. Second equality follows from the fact that the distribution of all points of a Poisson point process are . Since Y is a Bernoulli process independent of the underlying process S with \E[Y(S_i)] = p(S_i), we get From the distribution \frac{\Lambda'(x)}{\Lambda(A)} for x \in S \cap A for the Poisson point process S, we get Result follows from taking increasing sequences of sets A_k \uparrow \sX and monotone convergence theorem. ◻