Introduction
In this part we will introduce more properties of Poisson processes and Spatial Poisson processes.
Superposition and Thinning of Poisson Processes
Lemma: Superposition of Poisson Processes
Let $\{N_t^{(1)}\}_{t \geq 0}, \ldots, \{N_t^{(n)}\}_{t \geq 0}$ be independent Poisson processes with parameters $\lambda p_1, \ldots, \lambda p_n$, respectively, where $p = (p_1, \ldots, p_n)$ is a probability vector. If $c = (c_1, \ldots, c_n)$ are the counts after time $t$ (i.e., $c_i = N_t^{(i)}$), then, the conditional distribution of $(N_t^{(1)}, \ldots, N_t^{(n)})$, an equivalent model is, $$ c \sim \mathrm{Multinomial}(N_t, p), $$ where $\{N_t\}_{t \geq 0}$ is a Poisson process with parameter $\lambda$.
Proof
Using the model with independent Poisson processes, the probability of observing the count vector $c$ after time $t$ is (denoting $N = c_1 + \ldots + c_n$), $$ \begin{align*} \prod_{i = 1}^n \mathrm{Poisson}(c_i; \lambda p_i t) & = \prod_{i = 1}^n e^{-\lambda p_i t} \frac{(\lambda p_i t)^{c_i}}{c_i!} \newline & = e^{-\lambda t} (\lambda t)^N \prod_{i = 1}^n \frac{p_i^{c_i}}{c_i!} \newline & = e^{-\lambda t} \frac{(\lambda t)^N}{N!} \cdot \frac{N!}{c_1! c_2! \ldots c_n!} p_1^{c_1} p_2^{c_2} \ldots p_n^{c_n} \newline & = \mathrm{Poisson}(N; \lambda t) \cdot \mathrm{Multinomial}(c; N, p) \newline \end{align*} $$ The process for $N$ inherits independent and stationary increments from the sub-processes, so it follows it also a Poisson process with parameter $\lambda$.
Uniformly Distributed Arrival Times
Lemma: Uniformly Distributed Arrival Times
Let $\{N_t\}_{t \geq 0}$ be a Poisson process with parameter $\lambda$. If we fix that $N_t = k$, and we select uniformly randomly one of the $k$ arrivals, then its arrival time is uniformly distributed on the interval $[0, t]$.
Proof
Let $S_1, S_2, \ldots, S_k$ be the arrival times given that $N_t = k$. $$ \begin{align*} P(S_k \geq s \mid k \text{ uniformly random in } {1, \ldots, n}, N_t = n) & = \frac{1}{n} \sum_{k = 1}^{n} P(S_k \geq s \mid N_t = n) \newline & = \frac{1}{n} \sum_{k = 1}^{n} \sum_{j = 0}^{k - 1} P(N_s = j \mid N_t = n) \newline & = \frac{1}{n} \sum_{k = 1}^{n} \sum_{j = 0}^{k - 1} \frac{P(N_s = j) P(N_{t - s} = n - j)}{P(N_t = n)} \newline & = \frac{1}{n} \sum_{k = 1}^{n} \sum_{j = 0}^{k - 1} \frac{e^{-\lambda s} \frac{(\lambda s)^j}{j!} e^{-\lambda (t - s)} \frac{(\lambda (t - s))^{n - j}}{(n - j)!}}{e^{-\lambda t} \frac{(\lambda t)^n}{n!}} \newline & = \frac{1}{n} \sum_{j = 0}^{n - 1} (n - j) \frac{n!}{j! (n - j)!} \left(\frac{s}{t}\right)^j \left(1 - \frac{s}{t}\right)^{n - j} \newline & = \left[\sum_{j = 0}^{n - 1} \frac{n!}{j! (n - j - 1)!} \left(\frac{s}{t}\right)^j \left(1 - \frac{s}{t}\right)^{n - j - 1}\right] \cdot \left(1 - \frac{s}{t}\right) \newline & = 1 - \frac{s}{t} \end{align*} $$
Spatial Poisson Processes and Non-Homogeneous Poisson Processes
Definition: Spatial Poisson Process
A collection of random variables $\{N_A\}_{A \subseteq \mathbb{R}^d}$ is a spatial Poisson process with parameter $\lambda$ if,
- For each bounded set $A \subseteq \mathbb{R}^d$, $N_A$ has a Poisson distribution with parameter $\lambda |A|$.
- Whenever $A \subseteq B, N_A \leq N_B$ (i.e., spatial counting process).
- Whenever $A$ and $B$ are disjoint, $N_A$ and $N_B$ are independent.
Definition: Non-Homogeneous Poisson Process
A counting process $\{N_t\}_{t \geq 0}$ is a non-homogeneous Poisson process with intensity function $\lambda(t)$ if,
- $N_0 = 0$.
- For $0 < s < t$, $$ N_t - N_s \sim \mathrm{Poisson}\left(\int_{s}^{t} \lambda(x) \ dx\right). $$
- It has independent increments.