Nous présentons de nouveaux résultats d’approximation par des processus de Poisson pour les fonctionnelles stabilisantes de processus de Poisson et binomiaux. Ces fonctionnelles peuvent avoir des interactions non bornées, et recouvrent de nombreux exemples en géométrie stochastique. Nous obtenons des bornes pour la distance de Kantorovich–Rubinstein grâce par la méthode de Stein pour le générateur. Nous donnons différents types de bornes pour différents processus ponctuels. Alors que certaines de nos bornes soient données en termes de couplage du processus ponctuel avec sa version Palm, les autres sont exprimées en termes de la structure de dépendance locale, formalisée par la notion de stabilisation. Nous donnons deux exemples significatifs de ce nouveau cadre – l’un pour les points critiques Morse de la fonction distance, l’autre pour les grandes boules pour les voisins les plus proches. Nos bornes étendent considérablement les résultats de Barbour et Brown (1992), Decreusefond, Schulte et Thäle (2016) et Otto (2020).
We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich–Rubinstein distance using the generator approach to Stein’s method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework – one is for Morse critical points of the distance function, and the other is for large -nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thäle (2016) and Otto (2020).
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Mots clés : Functional limit theorems, Poisson process approximation, Kantorovich-Rubinstein distance, Point processes, Stein’s method, Glauber dynamics, Palm coupling, Stabilizing statistics, $k$-nearest neighbor balls, Morse critical points, Binomial point processes
@article{AHL_2022__5__1489_0, author = {Bobrowski, Omer and Schulte, Matthias and Yogeshwaran, D.}, title = {Poisson process approximation under stabilization and {Palm} coupling}, journal = {Annales Henri Lebesgue}, pages = {1489--1534}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.156}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ahl.156/} }
TY - JOUR AU - Bobrowski, Omer AU - Schulte, Matthias AU - Yogeshwaran, D. TI - Poisson process approximation under stabilization and Palm coupling JO - Annales Henri Lebesgue PY - 2022 SP - 1489 EP - 1534 VL - 5 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.156/ DO - 10.5802/ahl.156 LA - en ID - AHL_2022__5__1489_0 ER -
%0 Journal Article %A Bobrowski, Omer %A Schulte, Matthias %A Yogeshwaran, D. %T Poisson process approximation under stabilization and Palm coupling %J Annales Henri Lebesgue %D 2022 %P 1489-1534 %V 5 %I ÉNS Rennes %U http://www.numdam.org/articles/10.5802/ahl.156/ %R 10.5802/ahl.156 %G en %F AHL_2022__5__1489_0
Bobrowski, Omer; Schulte, Matthias; Yogeshwaran, D. Poisson process approximation under stabilization and Palm coupling. Annales Henri Lebesgue, Tome 5 (2022), pp. 1489-1534. doi : 10.5802/ahl.156. http://www.numdam.org/articles/10.5802/ahl.156/
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