Poisson (point) process is the simplest and most fundamental point process, which arose independently in multiple fields around 1909.

## Basic Definitions

An arrival process is a sequence of increasing positive r.v.’s, $( S_1, S_2, \cdots )$. And the r.v.’s $S_1, S_2, \cdots$ are called arrival epochs.

A renewal process is an arrival process for which the sequence of inter-arrival times is a sequence of IID r.v.’s.

A counting process is a continuous time stochastic process where the random variable at any given epoch represents the number of arrivals up to and including that epoch. $\{ N(t); t > 0 \}$

A counting process has stationary increments if for every $t' > t > 0$, $N(t') - N(t)$ has the same distribution function as $N(t' - t)$.

A counting process has independent increments if for every integer $k>0$ and every k-tuple of times $0 < t_1 < t_2 < \cdots < t_k$, the k-tuple of rv’s $N(t_1), \tilde{N}(t_1,t_2), \cdots ,\tilde{N}(t_{k-1},t_k)$ are statistically independent.

Two random processes are independent if any finite subsets of r.v.'s in these processes are independent from each other.

## Three Equivalent Definitions of Poisson Process

Although proof of equivalence is omitted, the following definitions of Poisson process are equivalent.

### Definition as an Arrival Process

A Poisson process is a renewal process with exponentially distributed inter-arrival intervals. The arrival rate of a Poisson process is $\lambda$, if the inter-arrival times $X \sim \text{Exp}(\lambda)$.

### Definition as a Counting Process

A Poisson process is a counting process that has independent and stationary increments, with Poisson distributed number of arrivals: $N(t) \sim \text{Poisson}(\lambda t), \forall t>0$.

### Definition as a limit of shrinking Bernoulli processes

A Poisson process is a counting process that has stationary and independent increments, and satisfies

1. $P( \tilde{N}(t,t+\delta) = 0 ) = 1 - \lambda \delta + o(\delta) )$
2. $P( \tilde{N}(t,t+\delta) = 1 ) = \lambda \delta + o(\delta) )$
3. $P( \tilde{N}(t,t+\delta) \leq 2 ) = o(\delta) )$

## Properties

Denoting Poisson process as an arrival process: $\mathbf{X} = (X_{(1)}, X_{(2)}, \cdots)$

Complete independence: Subprocesses of a Poisson process on disjoint domains are mutually independent random processes.

$$\forall B_1, B_2 \in [0,+\infty), B_1 \cap B_2 = \emptyset : \mathbf{X} \cap B_1 \perp \mathbf{X} \cap B_1$$

The next occurrence epoch in a Poisson process, regardless of occurrence at and before the current epoch, is exponentially distributed with occurrence rate λ. So is any inter-arrival time.

$$X_{(1)}, X_{(n+1)} - X_{(n)} \sim \text{Exp}(\lambda)$$

The occurrence epoch of a certain order in a Poisson process, regardless of occurrence at and before the current epoch, is Gamma distributed with parameters being occurrence rate λ and ordinal number k. So is the time between a certain number of arrivals.

$$X_{(k)}, X_{(n+k)} - X_{(n)} \sim \Gamma(\lambda,k)$$

The number of occurrences during any time interval in a Poisson process is Poisson distributed with expected occurrence λt.

$$K_t \equiv |\mathbf{X} \cap (t_0, t_0 +t]| \sim \text{Poisson}(\lambda t)$$

Conditional on the number of occurrences in a certain time interval, the distribution of occurrences in a homogeneous Poisson process is uniform over the time interval.

$$\mathbf{X} | K_t \sim U(0,t)^{K_t}$$

Inspection paradox: For any renewal process, given the renewal time is larger than certain value, the total renewal interval is stochastically larger than an unconditioned renewal interval. $$\forall t > 0, x > 0, P(S_{X_t + 1} > x) \ge P(S_1 > x)$$, or equivalently, $$\forall t > 0, x > 0, P(S > x | S > t) \ge P(S > x)$$. In particular, for Poisson processes $P(S > x | S > t) = P(S > x - t) > P(S > x)$. This means if a public transit system is not perfectly on schedule, the population average waiting time will be larger than half of the scheduled inter-arrival time.

## Operations

### Thinning

Independent thinning results in another Poisson process.

If a Poisson process is split by a Bernoulli process, the resulting processes are independent Poisson processes, with rates $\lambda_1 = p \lambda$ and $\lambda_2 = (1-p) \lambda$ respectively. Conditional on the original process, however, the two new processes are complement to each other.

A p(x)-thinning operation applied to a Poisson process with intensity measure Λ gives a Poisson process of removed points with intensity measure $\Lambda_p$: for any bounded Borel set B,

$$\Lambda_p(B)= \int_B p(x) \Lambda(dx)$$

### Superposition

The sum process of two independent Poisson processes of rates $\lambda_1$ and $\lambda_2$ is a Poisson process of rate $\lambda_1 + \lambda_2$.

Superposition theorem: The superposition of independent Poisson processes with intensity measures $\Lambda_1,\Lambda_2,\dots$ is a Poisson process with intensity measure

$$\Lambda = \sum \limits_{i=1}^{\infty} \Lambda_i$$

### Clustering

The clustering operation is replacing each point of some point process by another (possibly different) point process.

### Random Displacement

Displacement theorem: A Poisson process on $\mathbb{R}^d$ with intensity measure $\lambda(x)$, after independent random transformation, or displacement, with probability density $\rho(x,\cdot)$, is a Poisson process with intensity measure:

$$\lambda'(y) = \int_{\mathbb{R}^d} \lambda(x) \rho(x,y) dx$$

### Mapping

Mapping theorem: A Poisson process on $\mathbb{R}^d$ with intensity measure $\lambda(x)$, after transformation $f: \mathbb{R}^d \rightarrow \mathbb{R}^{d'}$, is a Poisson point process with intensity measure: for any bounded Borel set B on $\mathbb{R}^{d'}$,

$$\Lambda'(B)=\Lambda(f^{-1}(B))$$

## Non-homogeneous Poisson Process

A non-homogeneous Poisson process with time varying arrival rate $\lambda(t)$ is a counting process that has independent increments, and satisfies

1. $P( \tilde{N}(t,t+\delta) = 0 ) = 1 - \lambda(t) \delta + o(\delta) )$
2. $P( \tilde{N}(t,t+\delta) = 1 ) = \lambda(t) \delta + o(\delta) )$
3. $P( \tilde{N}(t,t+\delta) \leq 2 ) = o(\delta) )$

A non-homogeneous Poisson process could be viewed as a homogeneous Poisson process on a nonlinear time scale.