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

2025/11/13 by Hlyniana, Kateryna
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2511.10083

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 ≤ 2r and of order |W| λ² r^d; and (iii) a Stein bound in the Barbour--Brown d₂ metric controlled by ∫_{‖h‖≤ 2r} |g(h)-1| dh.

Citations

• Generalizations of Matérn's hard-core point processes
• Poisson process Fock space representation, chaos expansion and covariance inequalities
• Distance estimates for dependent thinnings of point processes with densities

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