Tag: Random measure

Criteria for Poisson Point Processes

If S is a finite random set in a standard Borel measurable space {(E,\mathcal E)} satisfying the following two properties,

  • if {A,B\in\mathcal E} are disjoint, then the sizes of {S\cap A} and {S\cap B} are independent random variables,
  • {{\mathbb P}(x\in S)=0} for each {x\in E},

then it is a Poisson point process. That is, the size of {S\cap A} is a Poisson random variable for each {A\in\mathcal E}. This justifies the use of Poisson point processes in many different areas of probability and stochastic calculus, and provides a convenient method of showing that point processes are indeed Poisson. If the theorem applies, so that we have a Poisson point process, then we just need to compute the intensity measure to fully determine its distribution. The result above was mentioned in the previous post, but I give a precise statement and proof here. Continue reading “Criteria for Poisson Point Processes” →

Poisson Point Processes

bomb map
Figure 1: Bomb map of the London Blitz, 7 October 1940 to 6 June 1941.
Obtained from http://www.bombsight.org (version 1) on 26 October 2020.

The Poisson distribution models numbers of events that occur in a specific period of time given that, at each instant, whether an event occurs or not is independent of what happens at all other times. Examples which are sometimes cited as candidates for the Poisson distribution include the number of phone calls handled by a telephone exchange on a given day, the number of decays of a radio-active material, and the number of bombs landing in a given area during the London Blitz of 1940-41. The Poisson process counts events which occur according to such distributions.

More generally, the events under consideration need not just happen at specific times, but also at specific locations in a space E. Here, E can represent an actual geometric space in which the events occur, such as the spacial distribution of bombs dropped during the Blitz shown in figure 1, but can also represent other quantities associated with the events. In this example, E could represent the 2-dimensional map of London, or could include both space and time so that {E=F\times{\mathbb R}} where, now, F represents the 2-dimensional map and E is used to record both time and location of the bombs. A Poisson point process is a random set of points in E, such that the number that lie within any measurable subset is Poisson distributed. The aim of this post is to introduce Poisson point processes together with the mathematical machinery to handle such random sets.

The choice of distribution is not arbitrary. Rather, it is a result of the independence of the number of events in each region of the space which leads to the Poisson measure, much like the central limit theorem leads to the ubiquity of the normal distribution for continuous random variables and of Brownian motion for continuous stochastic processes. A random finite subset S of a reasonably ‘nice’ (standard Borel) space E is a Poisson point process so long as it satisfies the properties,

  • If {A_1,\ldots,A_n} are pairwise-disjoint measurable subsets of E, then the sizes of {S\cap A_1,\ldots,S\cap A_n} are independent.
  • Individual points of the space each have zero probability of being in S. That is, {{\mathbb P}(x\in S)=0} for each {x\in E}.

The proof of this important result will be given in a later post.

We have come across Poisson point processes previously in my stochastic calculus notes. Specifically, suppose that X is a cadlag {{\mathbb R}^d}-valued stochastic process with independent increments, and which is continuous in probability. Then, the set of points {(t,\Delta X_t)} over times t for which the jump {\Delta X} is nonzero gives a Poisson point process on {{\mathbb R}_+\times{\mathbb R}^d}. See lemma 4 of the post on processes with independent increments, which corresponds precisely to definition 5 given below. Continue reading “Poisson Point Processes” →