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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?

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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. –  Dilip Sarwate Dec 17 '11 at 2:58
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Where did you find this definition? Are you quoting everything verbatim? –  Michael Hardy Dec 17 '11 at 3:03
    
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 $\lambda$. –  Dilip Sarwate Dec 17 '11 at 3:04
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1 Answer

This definition is more general, in that it characterizes both spatial and temporal Poisson processes. Its elements are the following:

  1. A partition $\cal{A}$ of $\mathbb{R}^d$ into measurable sets of finite measure. Carving $\mathbb{R}^d$ into finite-sized zones is done so that the process on the infinite region can be defined as a product of independent processes on finite regions. The actual partition is irrelevant; the same process is defined for any such $\cal{A}$.
  2. Independent random variables $Y_A ∼ \text{Poisson}(\lambda l(A))$ for each $A \in \cal{A}$. The variable $Y_A$ is the number of events occurring in region $A$.
  3. Independent random variables $U_{A,j} ∼ \text{Uniform}(A)$ for each $A \in \cal{A}$ and $j\in\{1,2,3,...\}$. The variable $U_{A,j}$ is the position of the $j$-th event in region $A$, if there is one.
  4. The (a.s. infinite) set of points $S=\bigcup_{A\in \cal{A}}\bigcup_{j\le Y_A}\left\{U_{A,j}\right\}$. Within each region $A$, the number of events is Poisson-distributed; and given the number of events, the locations of the events are uniformly distributed.

Taking $d=1$ and interpreting $\mathbb{R}$ as time gives a standard temporal Poisson process.

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