Difficulty in understanding this definition of Poisson process 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 $\lambda >0$.

*

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


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


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


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


*

*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 $\cal{A}$.

*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 $A$.

*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.

*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.
