The law of thin processes: a law of large numbers for point processes

TL;DR

This paper proves that superimposing many identical point processes and thinning them results in a Poisson process, illustrating a law of large numbers analogy for point processes.

Contribution

It provides a simple proof of a law of large numbers for point processes, connecting superposition, thinning, and Poisson convergence.

Findings

Superposition of IID point processes converges to Poisson process

Thinning by 1/n leads to Poisson process as n increases

Highlights analogy to classical laws of large numbers

Abstract

If you take a superposition of n IID copies of a point process and thin that by a factor of 1/n, then the resulting process tends to a Poisson point process as n tends to infinity. We give a simple proof of this result that highlights its similarity to the law of large numbers and to the law of thin numbers of Harremo\"es et al.