Neighbour-count dependent thinning of Poisson processes: correlation structure and Poisson approximation

TL;DR

This paper analyzes a local thinning process of Poisson point processes, deriving explicit formulas for correlation structures and approximations, revealing how neighbor-dependent retention probabilities influence clustering or inhibition.

Contribution

It provides exact intensity formulas, pair correlation functions, and Poisson approximations for neighbor-dependent thinning of Poisson processes, extending understanding of spatial point process interactions.

Findings

Derived exact intensity via Poisson--mixture formula.

Established pair correlation based on three-region overlap.

Provided bounds for Poisson process approximation.

Abstract

We study a local thinning $T_r$ that retains a point with probability $p(n_r)$, where $n_r$ counts neighbors within radius $r$. For Poisson input with spatially varying intensity, we obtain an exact intensity via a Poisson--mixture formula and a small-radius expansion. For homogeneous input we give a closed-form pair correlation based on the three-region overlap . First-order contact-scale asymptotics identify how the values $p(0),p(1),p(2)$ govern inhibition or clustering. On bounded windows we approximate $T_r(X)$ by a Poisson process with matched intensity through three routes: (i) a direct coupling to an independent thinning giving a total-variation bound; (ii) a Laplace-functional error supported at distances $\le 2r$ and of order $|W|\,\lambda^2 r^d$; and (iii) a Stein bound in the Barbour--Brown $d_2$ metric controlled by $\int_{\|h\|\le 2r} |g(h)-1|\,dh$.