Point process

A point process records random locations by a random measure that counts them. Examples include arrival times of a queue, positions of stars, and the finite eigenvalue configuration of a random matrix. The count in an observation window is a random variable; its expected value is supplied by the intensity measure of a point process, while its fluctuations and relationships with other windows require more information.

For a precise spatial convention, take $E={\mathbb{R}}^d$ with its Borel sigma-algebra and a probability space $(\Omega ,\mathcal{F},\mathbb{P})$. A locally finite point process $\Pi$ is a measurable function from $\Omega$ to the space ${\mathcal{N}}_{\mathrm{lf}}(E)$ of locally finite point measures, equipped with the sigma-algebra generated by evaluations $\nu \mapsto \nu (A)$ for Borel sets $A$. Each realization has the formwhere $J(\omega )$ is finite or countable and repeated locations are allowed; ${\delta }_x$ is a Dirac measure. Here a point measure is a counting measure with possible multiplicities, rather than the counting measure of a set of distinct locations.^[ref]

Local finiteness means ${\Pi }_{\omega }(B)<\infty$ for every bounded Borel set $B$, on a single almost sure event. In ${\mathbb{R}}^d$ it suffices to check the countable collection of balls of integer radius centered at the origin, and this is equivalent to finiteness on every compact set. On a general measurable space there is no notion of boundedness without additional structure: state a metric or an explicit family of observation windows. Another convention on locally compact spaces requires finiteness on compact sets; this need not coincide with finiteness on all bounded sets outside ${\mathbb{R}}^d$. Local finiteness does not require finitely many points in all of $E$.^[ref]

A point process is simple when, on a single almost sure event, every location has multiplicity at most one. Its realizations are simple locally finite point measures. For instance, ${\delta }_{0.4}+2{\delta }_{1.1}+{\delta }_{1.8}$ is locally finite but not simple: it has four points counted with multiplicity at three distinct locations. Simplicity is a statement about each entire realization. Checking $\mathbb{P}(\Pi (\{x\})>1)=0$ separately for every fixed $x$ is insufficient: $2{\delta }_U$ with $U$ having the uniform distribution on $(0,1)$ passes every such fixed-location check, yet always has a double point.^[ref]

For a bounded Borel set $W\subset {\mathbb{R}}^d$, the window count $N_W$ of a locally finite point process is a finite nonnegative integer random variable. It counts multiplicities, and the convention for window boundaries matters. On the time half-line, $N(t)=\Pi ((0,t])$ is a counting process and $N(t)-N(s)=\Pi ((s,t])$. For pairwise disjoint observation windows $W_1,\dots ,W_m$, the measure identityis valid for every realization. Additivity of counts supplies no independence assumption.

Observation windows are measurable sets used to count points of a point process. They are pairwise disjoint when $W_i\cap W_j=\varnothing$ for every $i\ne j$. Half-open intervals such as $[0,1)$ and $[1,2)$ meet at a boundary without sharing any point, whereas the closed intervals $[0,1]$ and $[1,2]$ both contain one. A disjoint union makes window counts additive; it does not make their random variables independent.

A point process on ${\mathbb{R}}^d$ is stationary if every translation preserves its probability law. If its mean count in a unit cube is finite, its intensity measure is $\mu (A)=\rho |A|$ for some $\rho \ge 0$, where $|A|$ denotes Lebesgue measure. This follows because stationarity makes the mean measure invariant under translations. A constant intensity function of a point process describes only these mean counts, and by itself asserts neither stationarity of the entire law nor independence.^[ref]

A Neyman-Scott process places independent offspring clusters around points of a Poisson parent process, with an independently drawn offspring count and displacements. Poisson offspring counts make the offspring process conditionally Poisson given the parents, and therefore give a Cox process. Arbitrary offspring counts need not have that property.

The intensity measure of a point process $\Pi$ is its expected counting measure:It may take the value $+\infty$.^[ref] Countable additivity follows by applying the monotone convergence theorem to the counts of finite unions of pairwise disjoint observation windows. If $\mu$ is locally finite and absolutely continuous with respect to Lebesgue measure, its Radon-Nikodym derivative is the intensity function of a point process $\lambda$:This is an expected number per unit volume, not a probability density function for a randomly selected point. For example, a single deterministic point has $\mu ={\delta }_0$, a Dirac measure, and no Lebesgue measure density. The density representation requires its own assumption.^[ref]

For a bounded Borel set $W$, $\mathbb{E}[N_W]=\mu (W)$, but local finiteness alone does not ensure that this expected value is finite. Conversely, $\mu (W)<\infty$ implies $N_W<\infty$ almost surely, even when $W$ is unbounded. A constant finite intensity function of a point process on all of ${\mathbb{R}}^d$ has infinite total intensity measure, although every bounded window has finite expected value.

Here is an explicit example even within simple point processes. Let $K$ be a positive integer with probability mass function $\mathbb{P}(K=k)=1/[k(k+1)]$. The telescoping series sums to one, $K$ is finite almost surely, but $\mathbb{E}[K]={\sum }_{k\ge 1}1/(k+1)=\infty$. Conditional on $K$, sample $K$ independent points with the uniform distribution on $(0,1)$. The resulting point process has finitely many distinct points almost surely, while $\mu ((0,1))=\infty$. Thus finite counts in each realization and finite mean counts are separate requirements; local finiteness of the intensity measure is an integrability assumption.

The same intensity measure can belong to very different point processes. A Poisson point process with a locally finite intensity measure $\mu$ adds two requirements: each finite-intensity window count has the Poisson distribution of mean $\mu (W)$, and counts in pairwise disjoint observation windows are mutually independent. These requirements determine its probability law; they do not follow from the formula for $\mu$.^[ref]

A randomly shifted lattice point process, a unit-rate Poisson point process, and the shared random rate gives dependent point counts construction all satisfy $\mu (A)=|A|$ on the real line. Nevertheless, their counts in $[0,1)$ have respectively variance zero, one, and two. Counts in the disjoint observation windows $[0,1/2)$ and $[1/2,1)$ have respectively covariance $-1/4$, zero, and $1/4$. These exact calculations show why measuring mean density cannot identify a spacing or statistical dependence model.

Let $U$ have the uniform distribution on $[0,1)$ and place a point at every $k+U$, $k\in \mathbb{Z}$. This is a simple point process, locally finite, and a stationary point process: a translation only changes the phase modulo one, preserving its uniform distribution. For every Borel set $A$, the Tonelli theorem givesIts consecutive points are exactly one unit apart, despite having the same intensity measure as a unit-rate Poisson point process. This is the one-dimensional translated-grid construction.^[ref]

Every half-open interval of length one contains exactly one point. For $A=[0,1/2)$ and $B=[1/2,1)$, $\Pi (A)+\Pi (B)=1$, each count has the Bernoulli distribution with parameter $1/2$, andMore generally, an interval of length $L=m+r$, with $m$ a nonnegative integer and $0\le r<1$, contains $m$ or $m+1$ points with probabilities $1-r$ and $r$. Its count variance is $r(1-r)$, while a unit-rate Poisson point process has count variance $L$.

Draw one random variable $R$, taking values zero and two with equal probabilities. Conditional on $R$, generate a Poisson point process on the real line with constant rate $R$. This is a Cox process directed by the random measure $Rdx$: the same rate is shared by every window. It is a locally finite simple point process and a stationary point process. The law of total expectation gives $\mu (A)=\mathbb{E}[R]|A|=|A|$; the law of total variance and law of total covariance give, for bounded Borel sets $A,B$ with $A\cap B=\varnothing$,The conditional covariance vanishes by independence, but the shared conditional means fluctuate together. This distinguishes conditional from unconditional independence.^[ref]

For a window of length $L$, the exact probability mass function isIn particular, its void probability functional is $(1+e^{-2L})/2$, compared with $e^{-L}$ for the unit-rate Poisson point process. Whole realizations can be empty or comparatively crowded, while their mean density remains one. This example demonstrates shared-rate statistical dependence, rather than a model of short-range offspring clusters.

The void probability functional assigns to a measurable set the probability that it contains no atom of a random point measure. For a Poisson random measure of intensity $\mu$, it is $e^{-\mu (A)}$. The exponential void probabilities characterize a Poisson random measure result uses simplicity; doubling every atom of a Poisson measure preserves void probabilities but changes count distributions.

If bounded intervals have void probabilities $e^{-\lambda (I)}$ for Lebesgue measure $\lambda$, a fixed deterministic point $x$ has zero chance of being an atom. Surround it by intervals of lengths tending to zero and use $\mathbb{P}(M(\{x\})>0)\le 1-e^{-\lambda (I)}$. This holds for each fixed endpoint, not simultaneously for all real points of a nonempty point process.

A point measure is a countable sum of Dirac measures. It is simple if all its atoms have mass one, and locally finite if every bounded set has finite mass. Simplicity lets fine enough cells count individual points, whereas occupation indicators would lose multiplicities in a nonsimple measure.

For a fixed bounded interval with no atoms at its endpoints, the occupation count of fully contained half-open dyadic intervals increases to the true count of a simple locally finite point measure. Refinement cannot reduce occupancy. Finitely many distinct atoms in a bounding interval have positive minimum separation; for fine enough cells each atom lies alone, and every interior atom's cell is contained in the target interval. Thus the counts eventually agree, path by path.

A point process is determinantal with kernel $K$ relative to a specified reference measure if every correlation function of a point process is the indicated determinant. A rank-$m$ finite-rank projection kernel gives an exactly $m$-point process, whose symmetric joint density is $\det [K(x_i,x_j)]/m!$.

The $k$th correlation function is the density of the $k$th factorial moment measure: integrating a nonnegative test function against $R_k$ gives the expectation of its sum over ordered $k$-tuples of distinct points. For exactly $m$ exchangeably labelled points with symmetric joint density $f_m$, it is $m!/(m-k)!$ times the marginal density of $k$ labels. It is not normalized to have total mass one.