PreprintCC BY-NC-ND 4.0arXiv
A law of thin processes with neighbour-count thinning
Abstract
We consider the superposition Sn of n i.i.d. simple point processes on ℝd and apply a dependent thinning Tn, where the retention probability of a point depends on its local neighbour count within a radius rn. While superpositions under independent thinning converge to Poisson processes, we show that under a critical geometric scaling n vd rnd → τ∈ (0,∞), the local interactions are transformed in the limit into an inhomogeneous Poisson point process with modified intensity. We prove that the thinned sequence Tn(Sn) converges to a Poisson process with a non-linearly modified intensity λ̃(x) = λ(x)α(τλ(x)).
Versions
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BibTeX
@misc{hlyniana2026thin,
title = {{A law of thin processes with neighbour-count thinning}},
author = {Kateryna Hlyniana},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25733},
eprint = {2609.25733},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25733},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A law of thin processes with neighbour-count thinning
AU - Kateryna Hlyniana
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25733
UR - https://arxiv.org/abs/2609.25733
AB - We consider the superposition S_n of n i.i.d. simple point processes on ℝ^d and apply a dependent thinning T_n, where the retention probability of a point depends on its local neighbour count within a radius r_n. While superpositions under independent thinning converge to Poisson processes, we show that under a critical geometric scaling n v_d r_n^d → τ∈ (0,∞), the local interactions are transformed in the limit into an inhomogeneous Poisson point process with modified intensity. We prove that the thinned sequence T_n(S_n) converges to a Poisson process with a non-linearly modified intensity λ̃(x) = λ(x)α(τλ(x)).
ER -