Poisson Process Basics

Definition A counting process is a random process $\{N(t) : t \ge 0\}$ in continuous time in which $N(t)$ is the number of events, called arrivals, in the time interval $(0, t]$. Each $N(t)$ takes values in $\{0, 1, 2, \dots\}$, the process starts at $N(0) = 0$, and $N(s) \le N(t)$ whenever $s < t$. For $s < t$ the increment $N(t) - N(s)$ is the number of arrivals in the interval $(s, t]$. Each sample path is a nondecreasing step function that jumps at the arrival times.

Definition Let $\lambda > 0$. A counting process $\{N(t) : t \ge 0\}$ is a homogeneous Poisson process with rate $\lambda$ if the following hold.

1. The process starts at $N(0) = 0$.

2. For all $s \ge 0$ and $t > 0$, the count in $(s, s + t]$ has the $\operatorname{Pois}(\lambda t)$ distribution, so for $k = 0, 1, 2, \dots$

   $$\begin{aligned} &P\bigl(N(s + t) - N(s) = k\bigr) \\[0.4em] &\quad = \frac{e^{-\lambda t} (\lambda t)^{k}}{k!}. \end{aligned}$$

3. For every $k \ge 1$ and all times $0 \le t_0 < t_1 < \cdots < t_k$, the counts $N(t_1) - N(t_0), \dots, N(t_k) - N(t_{k-1})$ are mutually independent.

Homogeneous means that the rate is the same at every time. A Poisson distribution for the count in one interval does not by itself make a process a Poisson process, because the third condition, about how counts in different intervals relate, is part of the definition.

Definition The rate $\lambda$ is the expected number of arrivals per unit time, since $E[N(t)] = \lambda t$. It carries a unit, such as arrivals per minute. For an interval of length $t$ measured in the same time unit, the product $\lambda t$ is the Poisson parameter of the count, a pure number equal to the expected count. Changing the time unit changes $\lambda$ and $t$ together and leaves $\lambda t$ unchanged. The probability of at least one arrival in an interval of length $t$ is

which is less than $\lambda t$ for every $t > 0$ and close to $\lambda t$ when $\lambda t$ is small. The product $\lambda t$ is an expected count, not a probability, and it can exceed $1$.

Definition A counting process has stationary increments if for every $t > 0$ the distribution of $N(s + t) - N(s)$ is the same for all $s \ge 0$, so that it depends only on the length $t$ of the interval $(s, s + t]$. In a homogeneous Poisson process with rate $\lambda$ this common distribution is $\operatorname{Pois}(\lambda t)$. Stationarity is a statement about distributions only. It does not say that the counts in two windows of equal length are equal, and it says nothing about whether they are independent.

Definition A counting process has independent increments if for every $k \ge 1$ and all times $0 \le t_0 < t_1 < \cdots < t_k$ the counts $N(t_1) - N(t_0), \dots, N(t_k) - N(t_{k-1})$ are mutually independent. Equivalently, the counts in any finite collection of disjoint intervals of the form $(a, b]$ are mutually independent. Half-open intervals that share only an endpoint, such as $(0, 1]$ and $(1, 2]$, are disjoint.

Definition Counts over intervals that overlap share the arrivals in the overlap. For a homogeneous Poisson process with rate $\lambda > 0$ and times $0 < s < t$, the count $N(t)$ splits as

into independent counts over $(0, s]$ and $(s, t]$. The cumulative counts $N(s)$ and $N(t)$ themselves are dependent, since

More generally, the covariance of the counts in two intervals is $\lambda$ times the length of their overlap. Given $N(s) = j$, the later count is $j$ plus an independent $\operatorname{Pois}\bigl(\lambda(t - s)\bigr)$ count, so it cannot be less than $j$.