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

Kateryna Hlyniana

TL;DR

The paper analyzes an $r$-local neighbour-count thinning $T_r$ that retains a point with probability $p(n_r(x;X))$, deriving exact intensity formulas for Poisson and Cox inputs and an exact two-point structure for PPP inputs. It shows finite-range dependence with correlations determined by the overlap of two $r$-balls and provides explicit small-$r$ asymptotics that connect $p(0),p(1),p(2)$ to inhibition or clustering at contact scales. Three complementary routes—coupling with independent thinning (TV bound), Laplace-functional control, and Stein-based $d_2$ bounds—give Poisson-approximation guarantees on bounded windows, with explicit rates in small and moderate radius regimes. The results offer practical guidance on selecting retention rules to shape local interaction (attraction, repulsion) and on when Poisson approximations are effective, supported by precise quantitative bounds tied to short-range correlations. Overall, the work unifies thinning constructions under a rigorous framework that yields both exact structural results and usable approximation tools for spatial point-process modelling.

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 $|W|\,λ^2 r^d$; and (iii) a Stein bound in the Barbour--Brown $d_2$ metric controlled by $\int_{\|h\|\le 2r} |g(h)-1|\,dh$.

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

TL;DR

The paper analyzes an -local neighbour-count thinning that retains a point with probability , deriving exact intensity formulas for Poisson and Cox inputs and an exact two-point structure for PPP inputs. It shows finite-range dependence with correlations determined by the overlap of two -balls and provides explicit small- asymptotics that connect to inhibition or clustering at contact scales. Three complementary routes—coupling with independent thinning (TV bound), Laplace-functional control, and Stein-based bounds—give Poisson-approximation guarantees on bounded windows, with explicit rates in small and moderate radius regimes. The results offer practical guidance on selecting retention rules to shape local interaction (attraction, repulsion) and on when Poisson approximations are effective, supported by precise quantitative bounds tied to short-range correlations. Overall, the work unifies thinning constructions under a rigorous framework that yields both exact structural results and usable approximation tools for spatial point-process modelling.

Abstract

We study a local thinning that retains a point with probability , where counts neighbors within radius . 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 govern inhibition or clustering. On bounded windows we approximate 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 and of order ; and (iii) a Stein bound in the Barbour--Brown metric controlled by .

Paper Structure

This paper contains 14 sections, 22 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Neighbour-count dependent thinning
  4. Definition and examples
  5. Intensity formulas (Poisson and Cox input)
  6. Two–point correlation function for the transformation of Poisson input: exact formula and small–radius asymptotics
  7. Small–radius asymptotics
  8. Examples: attraction vs. repulsion at small scales
  9. Poisson approximation by complementary routes
  10. Poisson approximation using coupling and Laplace functional
  11. A Stein bound in terms of the short–range correlation
  12. Small–radius vregime and explicit rate
  13. Moderate radius regime
  14. Comparison of bounds and “best–case’’ retention rules

Key Result

Lemma 1

Fix $r>0$. On $(\mathbb{R}^d\times\mathbf N, \mathcal{B} \otimes \mathcal{N}),$ the map is measurable. Consequently, for any measurable $p:\mathbb N_0\to[0,1]$, the map $(x,\mathbf Y)\mapsto p (n_r(x;\mathbf Y))$ is measurable.

Theorems & Definitions (63)

  • Definition 1: Neighbour-count dependent thinning
  • Lemma 1: Joint measurability of the neighbour count
  • proof
  • Remark 1: Equivariance and stationarity
  • Example 1: Matérn type I
  • Example 2: Geometric soft-core family
  • Example 3: Count-favouring retention
  • Lemma 2: Intensity for inhomogeneous Poisson input
  • proof
  • Corollary 1: Small--$\mu_x(r)$ expansion with local remainder control
  • ...and 53 more