Deterministic Thinning of Finite Poisson Processes

TL;DR

This paper establishes a precise condition for coupling two finite Poisson processes so that one is a deterministic subset of the other, revealing surprising properties and asymptotic behavior.

Contribution

It provides a necessary and sufficient condition for deterministic thinning of finite Poisson processes, highlighting non-monotonicity and asymptotic thresholds.

Findings

Coupling exists if and only if a specific intensity condition is met.

Surprising lack of monotonicity in the coupling condition.

As intensities grow large, the coupling condition simplifies to a threshold on expected points.

Abstract

Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of Pi and Gamma such that Gamma is a deterministic function of Pi, and all points of Gamma are points of Pi. The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in Pi than in Gamma.