Conditional Poisson process approximation

TL;DR

This paper extends Stein's method to approximate conditional Poisson point processes, addressing the challenges of modeling rare events conditioned on at least one occurrence, with applications in spatial and temporal event analysis.

Contribution

It introduces a novel Stein's method framework for conditional Poisson point process approximation, utilizing an immigration-death process to derive bounds for Stein factors.

Findings

Developed Stein's method for conditional Poisson processes

Calculated bounds for Stein factors using immigration-death process

Addressed difficulties in modeling conditioned on at least one event

Abstract

Point processes are an essential tool when we are interested in where in time or space events occur. The basic starting point for point processes is usually the Poisson process. Over the years, Stein's method has been developed with a great deal of success for Poisson point process approximation. When studying rare events though, typically one only begins modelling after the occurrence of such an event. As a result, a point process that is conditional upon at least one atom, is arguably more appropriate in certain applications. In this paper, we develop Stein's method for conditional Poisson point process approximation, and closely examine what sort of difficulties that this conditioning entails. By utilising a characterising immigration-death process, we calculate bounds for the Stein factors.