2 Point Processes 2 Point Processes 2.2 The Poisson Process

2.1 Introduction

Background reading for this section is Diggle (2002) and Waller and Gotway (2004).

Definition 2.1.

A stochastic process is a collection of random variables $X_{i}$ where $i$ belongs to an indexing set, $I$ , and each $X_{i}$ takes values in a sample space $\Omega$ .

Example 2.1.

Time series provides a simple example of a stochastic process: $X_{1},X_{2},X_{3},\ldots$

Definition 2.2.

A spatial point process is a stochastic process, a realisation of which consists of a countable set of points $x_{i}$ in the plane. In this course, we usually have $\Omega=\mathbb{R}^{2}$ .

We often refer to these points as events to distinguish them from arbitrary points $x\in\mathbb{R}^{2}$ in the plane.

We write $N(A)$ for the number of events in a planar region $A$ ,

N(A)=\#({x}_{i}\in A).

Note that $N(A)$ is a random variable: different realisations of the process will result in both different numbers and locations of points.

Definition 2.3.

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The process is stationary if, for any integer $k$ and regions $A_{i}:i=1,\dots,k$ the joint distribution of $N(A_{1}),\dots,N(A_{k})$ is invariant to translation by an arbitrary amount ${x}$ .
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The process is isotropic if, for any integer $k$ and regions $A_{i}:i=1,\dots,k$ the joint distribution of $N(A_{1}),\dots,N(A_{k})$ is invariant to rotation through an arbitrary angle $\theta$ , i.e. no directional effects.
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The process is orderly if there can be no co-located obervations i.e.

$\lim_{|A|\rightarrow 0}\mathbb{P}[N(A)>1]=0.$