Stein's method and locally dependent point process approximation

TL;DR

This paper develops Stein's method for approximating point processes with local dependence, showing improved accuracy over traditional Poisson models especially as the mean number of events grows.

Contribution

It introduces a new family of approximating point processes and establishes Stein's method for both positive and negative dependence structures, overcoming limitations of compound Poisson approximations.

Findings

Error bounds decrease with increasing mean number of events

Method applies to both positively and negatively dependent structures

Examples demonstrate improved approximation accuracy

Abstract

Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson processes can provide adequate approximations. However, the accuracy of approximations does not improve or may even deteriorate when the mean number of events increases. In this paper, we investigate an alternative family of approximating point processes and establish Stein's method for their approximations. We prove two theorems to accommodate respectively the positively and negatively related dependent structures. Three examples are given to illustrate that our approach can circumvent the technical difficulties encountered in compound Poisson process approximation [see Barbour & M{\aa}nsson (2002)] and our approximation error bound decreases when the mean number of the random events increases, in contrast to increasing bounds for compound Poisson process approximation.