Point Processes — Mathematics Knowledge System

Overview

A point process is a random collection of points in a space (time, plane, or higher-dimensional spaces). The canonical example is the Poisson point process (PPP), where points occur independently with a given intensity measure. Point processes model events in time (arrivals, failures, neuron spikes), spatial patterns (galaxy distributions, cellular networks), and serve as building blocks for more complex stochastic models. Key characterisations use the void probabilities, intensity functions, and Palm theory (the distribution seen by a typical point).

Intuition

Imagine buses arriving at a stop. If arrivals are memoryless and independent, this is a Poisson process. A point process generalises this: the random set of arrival times is a random subset of R, characterised by how likely it is to have k arrivals in any interval. Beyond the Poisson process, one can have clustering (events come in bursts, like earthquakes with aftershocks) or regularity (events spread themselves out, like scheduled trains). Spatial point processes model locations of trees, stars, cell phone towers -- each is a random set of points in R^2.

Formal Definition

Definition

A point process on a Polish space S is a random variable N taking values in the set of locally finite counting measures on S. For each Borel set A, N(A) counts the random number of points in A. A Poisson point process with intensity measure Lambda has: (1) N(A) ~ Poisson(Lambda(A)) for each A, and (2) N(A_1), ..., N(A_k) are independent for disjoint A_i. The Laplace functional characterises the PPP: E[exp(-integral f dN)] = exp(-integral (1 - e^{-f}) dLambda) for non-negative f.

N(A) \sim \mathrm{Poisson}(\Lambda(A)) \text{ for each Borel } A

Poisson marginals

\mathcal{L}_N[f] = \mathbb{E}\!\left[e^{-\int f\, dN}\right] = \exp\!\left(-\int (1 - e^{-f(x)})\, \Lambda(dx)\right)

Laplace functional of PPP

\lambda(t) = \lim_{dt \to 0} \frac{P(N(t, t+dt] \ge 1)}{dt}

Intensity function

\rho(x_1, \ldots, x_k) = \lim_{|dx_i|\to 0} \frac{P(\text{point in each }dx_i)}{|dx_1|\cdots|dx_k|}

k-th order intensity (product density)

Notation

Notation	Meaning
$N(A)$	Number of points in set A
$\Lambda$	Intensity measure of point process
$\lambda(t)$	Intensity function (rate at time t)
$\rho^{(k)}$	k-th order product density

Theorems

Theorem 1: Superposition and Thinning of PPP

The superposition (union) of independent Poisson point processes with intensity measures Lambda_1, Lambda_2, ... is a PPP with intensity Lambda = Lambda_1 + Lambda_2 + ... Independent thinning of a PPP with intensity Lambda by a probability function p(x) gives a PPP with intensity p(x)*Lambda(dx). These operations preserve the PPP structure.

Theorem 2: Slivnyak-Mecke Theorem (Palm Theory)

For a Poisson point process N with intensity Lambda and a measurable function h(x, N): E[sum_{x in N} h(x, N)] = integral E[h(x, N union {x})] Lambda(dx). Equivalently, the Palm distribution of the PPP at a typical point x is the same as the original PPP union {x} -- the PPP has no interaction between points.

Theorem 3: Mapping Theorem

If N is a PPP on S with intensity Lambda and f: S -> T is a measurable map, then the image process f(N) = {f(x) : x in N} is a PPP on T with intensity measure f*(Lambda) (the pushforward). In particular, the image of a homogeneous PPP on R^d under a linear map is again a PPP.

Worked Examples

1
The event {R > r} occurs iff there are no PPP points in the disk B(0, r) of area pi*r^2.
$P(R > r) = P(N(B(0,r)) = 0) = e^{-\lambda \pi r^2}$
2
Therefore R has CDF F(r) = 1 - e^{-lambda*pi*r^2} (a Rayleigh distribution).
$f_R(r) = 2\lambda\pi r\, e^{-\lambda\pi r^2},\quad r > 0$
3
Mean nearest-neighbour distance: E[R] = 1/(2*sqrt(lambda)).
$\mathbb{E}[R] = \frac{1}{2\sqrt{\lambda}}$

✓ Answer

The nearest-neighbour distance has a Rayleigh distribution: P(R > r) = exp(-lambda*pi*r^2). Mean = 1/(2*sqrt(lambda)).

Practice Problems

Mediumfree response

State the Campbell formula and use it to compute the mean of the sum sum_{x in N} f(x) for a PPP with intensity Lambda.

Common Mistakes

Common Mistake

Assuming all point processes are Poisson.

The PPP is characterised by complete independence between points. Real-world point processes often have interaction: clustering (earthquakes and aftershocks), repulsion (packed atoms), or dependence (queuing networks). Hawkes processes, Cox processes, and determinantal point processes go beyond PPP.

Quiz

The superposition of two independent Poisson point processes with intensities lambda_1 and lambda_2 is:

A Cox processA renewal processA Poisson point process with intensity lambda_1 + lambda_2A cluster process

Historical Background

The mathematical theory of point processes was systematised by Daley and Vere-Jones in the 1970s-80s, building on earlier work by Khintchine (renewal processes, 1955), Palm (conditional distributions, 1943), and Ito (Poisson random measures, 1944). The PPP had appeared implicitly in physics (Poisson's work on rare events) and engineering (telephone traffic modeling by Erlang, 1909).

1909
Erlang models telephone traffic with Poisson processes
A.K. Erlang
1943
Palm introduces conditional intensities and Palm distributions
Conny Palm
1944
Ito introduces Poisson random measures for Levy processes
Kiyosi Ito
1988
Daley and Vere-Jones publish the definitive textbook on point processes
Daryl Daley, David Vere-Jones

Summary

A point process is a random counting measure; the PPP is the canonical example with independent increments.
PPP is characterised by Poisson marginals N(A) ~ Poisson(Lambda(A)) and independence on disjoint sets.
Campbell's formula: E[sum f(x)] = integral f Lambda(dx). Laplace functional characterises the PPP.
Palm theory (Slivnyak-Mecke): the PPP seen from a typical point is the PPP plus the extra point.

References

BookDaley, D.J. and Vere-Jones, D. An Introduction to the Theory of Point Processes. Springer, 2003.
BookKingman, J.F.C. Poisson Processes. Oxford, 1993.
WebsiteWikipedia -- Point process