Basic Theory

The Process

So far, we have studied the Poisson process as a model for random points in time. However there is also a Poisson model for random points in space. Some specific examples of such random points are

• Defects in a sheet of material.
• Raisins in a cake.
• Stars in the sky.

The Poisson process for random points in space can be defined in a very general setting. All that is really needed is a measure space $(S,\mathcal{S},\mu )$. So $S$ is a set (the underlying universal set for our random points), $\mathcal{S}$ is a $\sigma$-algebra of subsets of $S$ (as always, the allowable sets), and $\mu$ is a positive measure on $(S,\mathcal{S})$ (a measure of the size of sets). The most important special case is when $S$ is a (Lebesgue) measurable subset of ${\mathbb{R}}^d$ for some $d\in {\mathbb{N}}_{+}$, $\mathcal{S}$ is the $\sigma$-algebra of measurable subsets of $S$, and $\mu ={\lambda }_d$ is $d$-dimensional Lebesgue measure. Specializing further, recall the lower dimensional spaces:

1. When $d=1$, $S\subseteq \mathbb{R}$ and ${\lambda }_1$ is length measure.
2. When $d=2$, $S\subseteq {\mathbb{R}}^2$ and ${\lambda }_2$ is area measure.
3. When $d=3$, $S\subseteq {\mathbb{R}}^3$ and ${\lambda }_3$ is volume measure.

Of course, the characterizations of the Poisson process on $[0,\mathrm{\infty })$, in term of the inter-arrival times and the characterization in terms of the arrival times do not generalize because they depend critically on the order relation $\le$ on $[0,\mathrm{\infty })$. However the characterization in terms of the counting process generalizes perfectly to our new setting. Thus, consider a process that produces random points in $S$, and as usual, let $N(A)$ denote the number of random points in $A\in \mathcal{S}$. Thus $N$ is a random, counting measure on $(S,\mathcal{S})$

To draw parallels with the Poison process on $[0,\mathrm{\infty })$, note that part (a) is the generalization of stationary, Poisson-distributed increments, and part (b) is the generalization of independent increments. By convention, if $\mu (A)=0$ then $N(A)=0$ with probability 1, and if $\mu (A)=\mathrm{\infty }$ then $N(A)=\mathrm{\infty }$ with probability 1. (These distributions are considered degenerate members of the Poisson family.) On the other hand, note that if $0<\mu (A)<\mathrm{\infty }$ then $N(A)$ has a proper Poisson distribution on $\mathbb{N}$

In particular, $r$ can be interpreted as the expected density of the random points (that is, the expected number of points in a region of unit size), justifying the name of the parameter.

The Distribution of the Random Points

As before, the Poisson model defines the most random way to distribute points in space, in a certain sense. Assume that we have a Poisson process $N$ on $(S,\mathcal{S},\mu )$ with density parameter $r\in (0,\mathrm{\infty })$.

More generally, if $A$ contains $n$ points, then the positions of the points are independent and each is uniformly distributed in $A$.

So given $N(A)=n\in {\mathbb{N}}_{+}$, each of the $n$ random points falls into $B$, independently, with probability $p=\mu (B)/\mu (A)$, regardless of the density parameter $r$.

Thinning and Combining

Suppose that $N$ is a Poisson random process on $(S,\mathcal{S},\mu )$ with density parameter $r\in [0,\mathrm{\infty })$. Thinning (or splitting) this process works just like thinning the Poisson process on $[0,\mathrm{\infty })$. Specifically, suppose that the each random point, independently of the others is either type 1 with probability $p$ or type 0 with probability $1-p$, where $p\in (0,1)$ is a new parameter. Let $N_1$ and $N_0$ denote the random counting measures associated with the type 1 and type 0 points, respectively. That is, $N_i(A)$ is the number of type $i$ random points in $A$, for $A\in \mathcal{S}$ and $i\in \{0,1\}$.

This result extends naturally to $k\in {\mathbb{N}}_{+}$ types. As in the standard case, combining independent Poisson processes produces a new Poisson process, and the density parameters add.

Applications and Special Cases

Non-homogeneous Poisson Processes

A non-homogeneous Poisson process on $[0,\mathrm{\infty })$ can be thought of simply as a Poisson process on $[0,\mathrm{\infty })$ with respect to a measure that is not the standard Lebesgue measure ${\lambda }_1$ on $[0,\mathrm{\infty })$. Thus suppose that $r:[0,\mathrm{\infty })\to (0,\mathrm{\infty })$ is piece-wise continuous with ${\int }_0^{\mathrm{\infty }}r(t)dt=\mathrm{\infty }$, and let $$m(t)={\int }_0^{t}r(s)ds,t\in [0,\mathrm{\infty })$$ Consider the non-homogeneous Poisson process with rate function $r$ (and hence mean function $m$). Recall that the Lebesgue-Stieltjes measure on $[0,\mathrm{\infty })$ associated with $m$ (which we also denote by $m$) is defined by the condition $$m(a,b]=m(b)-m(a),a,b\in [0,\mathrm{\infty }),a<b$$ Equivalently, $m$ is the measure that is absolutely continuous with respect to ${\lambda }_1$, with density function $r$. That is, if $A$ is a measurable subset of $[0,\mathrm{\infty })$ then $$m(A)={\int }_{A}r(t)dt$$

Nearest Points in ${\mathbb{R}}^d$

In this subsection, we consider a rather specialized topic, but one that is fun and interesting. Consider the Poisson process on $({\mathbb{R}}^d,{\mathcal{R}}_d,{\lambda }_d)$ with density parameter $r>0$, where as usual, ${\mathcal{R}}_d$ is the $\sigma$-algebra of Lebesgue measurable subsets of ${\mathbb{R}}_d$, and ${\lambda }_d$ is $d$-dimensional Lebesgue measure. We use the usual Euclidean norm on ${\mathbb{R}}^d$: $$\parallel \mathit{x}{\parallel }_d={(x_1^d+x_2^d\cdots +x_d^d)}^{1/d},\mathit{x}=(x_1,x_2,\dots ,x_d)\in {\mathbb{R}}^d$$ For $t>0$, let $B_t=\{\mathit{x}\in {\mathbb{R}}^d:\parallel \mathit{x}{\parallel }_d\le t\}$ denote the ball of radius $t$ centered at the origin. Recall that ${\lambda }_d(B_t)=c_d{t}^d$ where $$c_d=\frac{{\pi }^{d/2}}{\mathrm{\Gamma }(d/2+1)}$$ is the measure of the unit ball in ${\mathbb{R}}^d$, and where $\mathrm{\Gamma }$ is the gamma function. Of course, $c_1=2$, $c_2=\pi$, $c_3=4\pi /3$.

For $t\ge 0$, let $M_t=N(B_t)$, the number of random points in the ball $B_t$, or equivalently, the number of random points within distance $t$ of the origin. From our formula for the measure of $B_t$ above, it follows that $M_t$ has the Poisson distribution with parameter $r{c}_d{t}^d$.

Now let $Z_0=0$ and for $n\in {\mathbb{N}}_{+}$ let $Z_n$ denote the distance of the $n$th closest random point to the origin. Note that $Z_n$ is analogous to the $n$th arrival time for the Poisson process on $[0,\mathrm{\infty })$. Clearly the processes $\mathit{M}=(M_t:t\ge 0)$ and $\mathit{Z}=(Z_0,Z_1,\dots )$ are inverses of each other in the sense that $Z_n\le t$ if and only if $M_t\ge n$. Both of these events mean that there are at least $n$ random points within distance $t$ of the origin.

Computational Exercises