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What is the difference between a poisson process and a poisson point process???? in terms of properties and definition ? If possible, please also provide an explanation in layman terms.

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At time t the former counts the points of the latter located in the interval (0,t). –  Did Dec 1 '12 at 11:03
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up vote 2 down vote accepted

A process $(N_t)_{t\geq 0}$ is called a Poisson process with parameter $\lambda>0$ if it is a Lévy process and

1) $N_t-N_s\sim\mathrm{po}(\lambda(t-s))$ for all $0\leq s<t$.

2) $t\mapsto N_t(\omega)$ is non-decreasing with values in $\mathbb{N}_0$ for almost all $\omega$.

Being a Lévy process means that $N_0=0$, the sample paths are càdlàg and that the process has independent increments. The sample paths looks something like this ($\lambda=1$):

When talking about Poisson point processes there is, first of all, no time parameter in question. Let's consider $\mathbb{R}^n$ and the family of all closed subsets $$ \mathcal{F}=\{F\subseteq \mathbb{R}^n\mid F\text{ is closed in }\mathbb{R}^n\}. $$ Now, one equips $\mathcal{F}$ with the topology of closed convergence and then we can talk about the Borel sets $\mathcal{B}(\mathcal{F})$ of $\mathcal{F}$. Let $(\Omega,\mathcal{A},P)$ be a probability space.

Definition: A mapping $Z:\Omega\to\mathcal{F}$ is called a random closed set if $Z$ is $(\mathcal{A},\mathcal{B}(\mathcal{F}))$-measurable. The measure $P\circ Z^{-1}$ is called the distribution of $Z$.

In order to introduce point processes in general, we need to introduce the family of all locally finite sets in $\mathbb{R}^n$: $$ \mathcal{F}_{\text{lf}}=\{F\in\mathcal{F}\mid \#(F\cap C)<\infty \text{ for all compact subsets }C\}. $$

Definition: A random closed set $X$ with $P_X(\mathcal{F}_{\text{lf}})=1$ is called a point process in $\mathbb{R}^n$. The function $\Theta$ on $\mathcal{B}(\mathbb{R}^n)$ given by $$ \Theta(A)=E[\#(X\cap A)],\quad A\in \mathcal{B}(\mathbb{R}^n) $$ is called the intensity measure of $X$.

With this in hand a Poisson point process is defined as:

Definition: Let $X$ be a point process on $\mathbb{R}^n$ with intensity measure $\Theta$. Then $X$ is called a Poisson point process if $\#(X\cap A)\sim \mathrm{po}(\Theta(A))$ for all $A$ with $\Theta(A)<\infty$.

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med = with? –  Did Dec 1 '12 at 11:04
    
@did: Thanks, a little Danish slipped in there :) –  Stefan Hansen Dec 1 '12 at 11:46
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Indeed. Chassez le naturel, il revient au galop, as they say. –  Did Dec 1 '12 at 12:23
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