4
$\begingroup$

I am having trouble in understanding this definition of Poisson process.

Let $S$ be a random discrete subset of points of $\mathbb{R}^d$ and let $\lambda >0$.

  1. A partition $\mathcal{A}$ of $\mathbb{R}^d$ with $A\in \mathcal{A}$ measurable and $l(A)<\infty$.

  2. Independent Poisson random variables $Y_A\sim\text{Poisson}(\lambda l(A))$.

  3. A family $((U_{A,j}, j\ge 1) A\in \mathcal{A})$, where $U_{A,j}\sim\text{Unif}(A)$ independent.

  4. Define $$S=\bigcup_{A\in \mathcal{A}}\bigcup_{j\le Y_A}\{U_{A,j}\}$$

$S$ is a Poisson process of intensity $\lambda$.

All I already knew was the definition given in the wikipedia page

Are these two different or have connection? Can some one help understanding this?

  • 2
    Sounds like a _spatial_ Poisson process rather than a _temporal_ Poisson process. In the latter, the number of arrivals in an interval $(t_1, t_2]$ of length $\tau = t_2-t_1$ is Poisson$(\lambda\tau)$ where $\lambda$ is the arrival rate; independent for non-overlapping Here we have the number of points in a set $A \in \mathbb R^d$ is Poisson$(\lambda l(A))$ where $l(A)$ is the measure of $A$; independent for disjoint subsets, and so on.2011-12-17
  • 1
    Where did you find this definition? Are you quoting everything verbatim?2011-12-17
  • 0
    And given the total number of points in $A$, the individual points are independent and uniformly distributed on $A$ which is analogous to a similar property enjoyed by temporal Poisson processes. Note that this does not depend on the value of $\lambda$.2011-12-17

1 Answers 1