Poisson Point Process

Picture a sample space \(\Omega\) (e.g. \(\mathbb{R}\)), with some intensity function \(f: \Omega \rightarrow \mathbb{R}_{\geq 0}\). For a given interval \(A \subset \Omega\), the random number of points that fall into the interval A is distributed Poisson with parameter equal to the integral of the intensity over the region:

\[N_A \sim Poisson(\int_{a \in A} f(a) da )\]

with the additional property that if two regions \(A, B\) are disjoint, then \(N_A\) and \(N_B\) are disjoint. This integral \(\int_A f(a) da\) can be viewed as a measure, which presents an equivalent view, that \(N(\cdot) \sim Poisson(F(\cdot))\) is a random measure.

Definitions

Counting Definition

A stochastic process \({N_t : t \geq 0}\) is said to be a Poisson point process if the process has three properties:

1. \[N_0 = 0\]
2. the process has independent increments, meaning \(\forall m \in \mathbb{N}\) and any choice \(t_0, t_1, ..., t_m\) with \(t_0 < t_1 < ... < t_m\), the following random variables are independent:

\[N_{t_1} - N_{t_0}, N_{t_2} - N_{t_1}, ..., N_{t_m} - N_{t_{m-1}}\]

1. The number of points in any interval of length \(t\) is distributed Poisson with rate \(\lambda t\).

Interarrival Definition

TODO

Point Process Definition

An alternative, equivalent definition is that the number of points on the real line in the interval \((a, b]\) is distributed Poisson with rate \((b-a) \lambda\) i.e.

\[P(N(a, b] = n) = \frac{(\lambda(b-a))^n e^{-\lambda(b-a)}}{n!}\]