Poisson Point Processes

Simple point processes

Consider the \(d\)-dimensional Euclidean space \(\R^d\). The collection of Borel measurable subsets \(\sB(\R^d)\) of the above Euclidean space is generated by sets \(B(x) \triangleq \set{y \in \R^d: y_i \le x_i}\) for \(x \in \R^d\).

Definition 1. A simple point process is a random countable collection of distinct points \(S: \Omega \to \sX^\N\), such that the distance \(\norm{S_n} \to \infty\) as \(n \to \infty\).

Remark 1. Since \(S\) is a simple point process, each point \(S_n\) is unique. Therefore, we can identify \(S\) as a random set of points in \(\sX\) and \(S\cap A\) is the random set of points in \(A\).

Remark 2. For any simple point process \(S\), we have \(P(\set{S_n = S_m \text{ for any } n \neq m}) = 0\) and \(\abs{S\cap A}\) is finite almost surely for any bounded set \(A \in \sB(\sX)\).

Example 2 (Simple point process on the half-line). We can simplify this definition for \(d=1\). When \(\sX = \R_+\), one can order the points of the process \(S: \Omega \to \R_+^\N\) to get ordered process \(\tilde{S}: \Omega \to \R_+^\N\), such that \(\tilde{S}_n = S_{(n)}\) is the \(n\)th order statistics of \(S\). That is, \(S_{(0)} \triangleq 0\), and \(S_{(n)} \triangleq \inf\set{S_k > S_{(n-1)}: k \in \N}.\) such that \(S_{(1)} < S_{(2)} < \dots < S_{(n)} < \dots\), and \(\lim_{n \in \N}S_{(n)} = \infty\). We will call this an arrival process.

Definition 3. Corresponding to a point process \(S:\Omega\to\sX^\N\), we denote the number of points in a set \(A \in \sB(\sX)\) by The resulting process \(N: \Omega \to {\Z_+}^ {\sB(\sX)}\) is called a counting process for the point process \(S: \Omega \to \sX^\N\).

Remark 3. Let \(A \in \sB(\sX)^k\) be a bounded partition of \(B\in \sB(\sX)\). From the disjointness of \((A_1, \dots, A_k)\), we have

Definition 4. A counting process is simple if the underlying point process is simple.

Remark 4. For a simple counting process \(N\), we have \(N(\set{x}) \le 1\) almost surely for all \(x \in \sX\).

Remark 5. Let \(N : \Omega \to {\Z_+}^{\sB(\sX)}\) be the counting process for the point process \(S: \Omega \to \sX^\N\).

1. Note that the point process \(S\) and the counting process \(N\) carry the same information.

2. The distribution of point process \(S\) is completely characterized by the finite dimensional distributions of random vectors \((N(A_1), \dots, N(A_k))\) for any bounded sets \(A_1, \dots, A_k \in \sB(\sX)\) and finite \(k \in \N\).

Example 5 (Simple point process on the half-line). Since the Borel measurable sets \(\sB(\R_+)\) are generated by half-open intervals \(\set{(0,t]: t \in \R_+}\), we denote the counting process by \(N: \Omega \to {\Z_+}^{\R_+}\), where \(N_t \triangleq N(0, t]= \sum_{n \in \N}\SetIn{S_n \in (0,t]}\) is the number of points in the half-open interval \((0,t]\). For \(s < t\), the number of points in interval \((s,t]\) is \(N(s,t] = N(0,t] - N(0,s] = N_t - N_s\).

Theorem 6 (R'{e}nyi). Distribution of a simple point process \(S:\Omega\to\sX^\N\) on a locally compact second countable space \(\sX\) is completely determined by void probabilities \((P\set{N(A)=0}: A \in \sB(\sX))\).

Proof. Proof. It suffices to show that the finite dimensional distributions of \(S\) on locally compact sets are characterized by void probabilities.

1. We will show this by induction on the number of points \(k\) in a bounded set \(A \in \sB\). Let \(A_1, \dots, A_k, B \in \sB(\sX)\) locally compact, then we will show that \(u_k \triangleq P(\cap_{i=1}^k\set{N(A_i)> 0}\cap\set{N(B) = 0})\) can be computed from void probabilities. From \(k=1\), we have The induction can be proved by the recursive relation

2. For any locally compact set \(B \in \sB(\sX)\), there exists a sequence of nested partitions \(B_n \triangleq (B_{n,j}: j \in [J_n])\) that eventually separates the points in \(S\cap B\) as \(n \to \infty\). We define the number of subsets of partition \((B_{n,j}: j \in [J_n])\) that consist of at least one point in \(S\cap B\), as \(H_n(B) \triangleq \sum_{j=1}^{J_n}\SetIn{N(B_{n,j}) > 0}\) where \(H_n(B) \uparrow N(B)\) almost surely.

3. We next show that for all locally compact sets \(B_1, \dots, B_k \in \sB(\sX)\) and \(j_1, \dots, j_k \in \N\), the probability \(P(\cap_{i=1}^k\set{H_n(B_i) = j_i})\) can be expressed in terms of void probabilities. We observe that This can be expressed in terms of void probabilities by Step 1.

4. For a simple point process, we have the following almost sure limit \(\lim_n\cap_{i=1}^k\set{H_n(B_i) = j_i} = \cap_{i=1}^k\set{N(B_i) = j_i}\). The result follows from the continuity of probability.

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Remark 6. Recall that \(\abs{A} = \int_{x \in A}dx\) is the volume of the set \(A \in \sB(\R^d)\) and for any such \(A\).

Definition 7. The intensity measure \(\Lambda:\sB(\sX)\to\R_+\) is defined for each bounded set \(A \in \sX\) as its scaled volume in terms of the intensity density \(\lambda: \R^d \to \R_+\), as If the intensity density \(\lambda(x) = \lambda\) for all \(x \in \R^d\), then \(\Lambda(A) = \lambda\abs{A}\). In particular for partition \(A_1, \dots, A_k\) for a set \(B\), we have \(\Lambda(B) = \sum_{i=1}^k\Lambda(A_i)\).

Poisson point process

Definition 8. A non-negative integer valued random variable \(N: \Omega \to \Z_+\) is called Poisson if for some constant \(\lambda > 0\), we have

Remark 7. It is easy to check that \(\E N = \Var[N] = \lambda\). Furthermore, the moment generating function \(M_{N_t} = \E e^{t N} = e^{\lambda(e^{t} - 1)}\) exists for all \(t \in \R\).

Corollary 9. A simple counting process \(N: \Omega \to \Z_+^{\sB(\sX)}\) has Poisson marginal distribution with intensity measure \(\Lambda:\sB(\sX)\to\R_+\) if and only if void probabilities are exponential with the same intensity measure \(\Lambda\).

Proof. Proof. It is clear that if the marginal distribution of the counting process \(N\) is Poisson with intensity measure \(\Lambda\), then the void probability \(P\set{N(A) = 0} = e^{-\Lambda(A)}\) is exponential for any bounded set \(A \in \sB(\sX)\).

Conversely, we assume that the void probabilities are exponentially distributed with intensity measure \(\Lambda\). It follows from the linearity of intensity measure that for any finite, bounded, and disjoint sets \(B_1, \dots, B_k \in \sB(\sX)\), we have That is, the Bernoulli random vector \((\SetIn{N(B_i) = 0}: i \in [k])\) is independent for any finite \(k \in \N\) and bounded disjoint \(\sB(\sX)\) measurable sets \(B_1, \dots, B_k\). Next we consider a set \(B \in \sB(\sX)\) and a partition \(B_n \triangleq (B_{n,j}: j \in [J_n])\) of \(B\) such that \(\Lambda(B_{n,j}) = \frac{\Lambda(B)}{J_n}\) for all \(j \in [J_n]\). It follows that \(H_n(B) \triangleq \sum_{j=1}^{J_n}\SetIn{N(B_{n,j})> 0}\) is the sum of \(J_n\) Bernoulli random variables with success probability \(p_n \triangleq 1-e^{-\Lambda(B)/J_n}\), and hence has a Binomial distribution with parameters \((J_n, p_n)\). Therefore, Recall that \(H_n(B) \uparrow N(B)\) as \(n\to\infty\) in the proof of Rényi’s Theorem, and \(\lim_{n\to\infty}J_n = \infty\) and \(\lim_{n\in\N}\abs{B_{n,j}}= 0\). Thus, \(\lim_{n\to\infty}\frac{J_n!}{(J_n-m)!}(e^{\Lambda(B)/J_n}-1)^m = \Lambda(B)^m\). Taking limit \(n\to\infty\) on both sides of the above equation, we get the result. ◻

Definition 10. A counting process \(N:\Omega\to\Z_+^{\sB(\sX)}\) has the completely independence property, if for any collection of finite disjoint and bounded sets \(A_1, \dots, A_k \in \sB(\sX)\), the vector \((N(A_1), \dots, N(A_k)):\Omega \to \Z_+^k\) is independent. That is,

Definition 11. A simple point process \(S: \Omega \to \sX^\N\) is Poisson point process, if the associated counting process \(N: \Omega \to \Z_+^{\sB(\sX)}\) has complete independence property and the marginal distributions are Poisson.

Definition 12. The intensity measure \(\Lambda: \sB(\sX) \to \R_+\) of Poisson process \(S\) is defined by \(\Lambda(A) \triangleq \E N(A)\) for all bounded \(A \in \sB(\sX)\).

Remark 8. Recall that for any partition \(A \in \sB(\sX)^k\) of a bounded set \(B \in \sB(\sX)\), we have \(N(B) = \sum_{i=1}^kN(A_i)\) and therefore it follows from the linearity of expectations that \(\Lambda(B) = \E N(B) = \sum_{i=1}^k\E N(A_i) = \sum_{i=1}^k\Lambda(A_i).\) Thus, this is a valid intensity measure.

Remark 9. For a Poisson process with intensity measure \(\Lambda\), it follows from the definition that for any finite \(k \in \Z_+\), and bounded mutually disjoint sets \(A_1, \dots, A_k \in \sB(\sX)\), we have

Definition 13. If the intensity measure \(\Lambda\) of a Poisson process \(S\) satisfies \(\Lambda(A) = \lambda\abs{A}\) for all bounded \(A \in \sB(\sX)\), then we call \(S\) a homogeneous Poisson point process and \(\lambda\) is its intensity.

Equivalent characterizations

Theorem 14 (Equivalences). Following are equivalent for a simple counting process \(N: \Omega \to {\Z_+}^{\sB(\sX)}\).

1. Process \(N\) is Poisson with locally finite intensity measure \(\Lambda\).

2. For each bounded \(A \in \sB(\sX)\), we have \(P\set{N(A) = 0} = e^{-\Lambda(A)}\).

3. For each bounded \(A \in \sB(\sX)\), the number of points \(N(A)\) is a Poisson with parameter \(\Lambda(A)\).

4. Process \(N\) has the completely independence property, and \(\E N(A) = \Lambda(A)\) for all bounded sets \(A \in \sB(\sX)\).

Proof. Proof. We will show that \(i\_ \implies ii\_ \implies iii\_ \implies iv\_ \implies i\_\).

1. It follows from the definition of Poisson point processes and definition of Poisson random variables.

2. From Corollary [cor:ExpPoisson], we know that if void probabilities are exponential, then the marginal distributions are Poisson.

3. We will show this in two steps.

   • Since the distribution of random variable \(N(A)\) is Poisson, it has mean \(\E N(A) = \Lambda(A)\).

   • Consider a partition \(A \in \sB^k\) for a bounded set \(B\in\sB(\sX)\), then \(\Lambda(B) = \Lambda(A_1) + \dots + \Lambda(A_k)\). Consider all partitions \(n\in\Z_+^k\) of a non-negative integer \(m \in \Z_+\), to write Using the definition of Poisson distribution, we can write the LHS of the above equation as Since the expansion of \((a_1 + \dots + a_k)^m = \sum_{n_1 + \dots+n_k = m}\binom{m}{n_1, \dots, n_k}\prod_{i=1}^ka_i^{n_i}\), we get Equating each term in the summation, we get \(P\set{N(A_1) = n_1, \dots, N(A_k) = n_k} = \prod_{i=1}^kP\set{N(A_i) = n_i}\).

4. From Corollary [cor:ExpPoisson], if the void probability is exponential with intensity measure \(\Lambda\), then the marginal distribution if Poisson with the same intensity measure. We define \(f:\sB(\sX)\to(-\infty, 0]\) by \(f(A) \triangleq \ln P\set{N(A) = 0}\) for all bounded \(A \in \sB(\sX)\). Then, we observe that for any partition \((A_1,\dots, A_k)\) of \(A\), we have \(f(\cup_{i=1}^kA_i) = \ln P\set{N(A)= 0} = \ln\prod_{i=1}^kP\set{N(A_i) = 0} = \sum_{i=1}^kf(A_i)\). It follows that \(-f:\sB(\sX)\to\R_+\) is an intensity measure, and \(P\set{N(A)=0} = e^{f(A)}\). Since \(\E N(A) = -f(A) = \Lambda(A)\), the result follows.

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Corollary 15 (Poisson process on the half-line). A random process \(N: \Omega \to \Z_+^{\R_+}\) indexed by time \(t \in \Z_+\) is the counting process associated with a one-dimensional Poisson process \(S: \Omega \to \R_+^\N\) having intensity measure \(\Lambda\) iff

1. Starting with \(N_0 = 0\), the process \(N_t\) takes a non-negative integer value for all \(t \in \R_+\);

2. the increment \(N_s - N_t\) is surely nonnegative for any \(s \ge t\);

3. the increments \(N_{t_1}, N_{t_2} - N_{t_1}, \dots , N_{t_n} - N_{t_{n-1}}\) are independent for any \(0 < t_1 < t_2 < \dots < t_{n-1} < t_n\);

4. the increment \(N_s - N_t\) is distributed as Poisson random variable with parameter \(\Lambda(t, s]\) for \(s \ge t\).

The Poisson process is homogeneous with intensity \(\lambda\), iff in addition to conditions \((a), (b), (c)\), the distribution of the increment \(N_{t+s} - N_t\) depends on the value \(s \in \R_+\) but is independent of \(t \in \R_+\). That, is the increments are stationary.

Proof. Proof. We have already seen that definition of Poisson processes implies all four conditions. Conditions \((a)\) and \((b)\) imply that \(N\) is a simple counting process on the half-line, condition \((c)\) is the complete independence property of the point process, and condition \((d)\) provides the intensity measure. The result follows from the equivalence \(iv\_\) in Theorem [thm:equiv]. ◻