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

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 $\mathrm{\mid }W\mathrm{\mid }{\lambda }^2{r}^d$; and (iii) a Stein bound in the Barbour–Brown $d_2$ metric controlled by ${\int }_{\mathrm{\parallel }h\mathrm{\parallel }\le 2r}\mathrm{\mid }g(h)-1\mathrm{\mid }dh$.