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. ◻