# Limit theory for geometric statistics of point processes having fast decay of correlations

1 DYOGENE - Dynamics of Geometric Networks
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique : UMR 8548, Inria de Paris
Abstract : Let $\mathcal{P}$ be a simple, stationary, clustering point process on $\mathbb{R}^d$ in the sense that its correlation functions factorize up to an additive error decaying exponentially fast with the separation distance. Let $\mathcal{P}_n:= \mathcal{P} \cap W_n$ be its restriction to windows $W_n := [- \frac{n^{1/d}}{2}, \frac{n^{1/d}}{2}]^d \subset \mathbb{R}^d$. We consider the statistic $H_n^\xi:= \sum_{x \in \mathcal{P}_n} \xi(x, \mathcal{P}_n)$ where $\xi(x, \mathcal{P}_n)$ denotes a score function representing the interaction of $x$ with respect to $\mathcal{P}_n$. When $\xi$ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for $H_n^\xi$ and, more generally, for statistics of the random measures $\mu_n^{\xi} := \sum_{x \in \mathcal{P}_n} \xi(x, \mathcal{P}_n) \delta_{n^{-1/d} x},$ as $W_n \uparrow \mathbb{R}^d$. This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the $k$-nearest neighbor graph) of determinantal point processes with fast decreasing kernels, including the $\alpha$-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov  to non-linear statistics. It also gives the limit theory for geometric U-statistics of permanental point processes as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin , which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [10,11] to show clustering of mixed moments of $\xi$. Clustering extends the cumulant method to the setting of purely atomic random measures, yielding the asymptotic normality of $\mu_n^{\xi}$.
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Journal articles
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https://hal.inria.fr/hal-01331939
Contributor : Bartlomiej Blaszczyszyn Connect in order to contact the contributor
Submitted on : Tuesday, June 14, 2016 - 7:04:33 PM
Last modification on : Wednesday, October 27, 2021 - 5:15:41 AM

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### Citation

Bartłomiej Błaszczyszyn, Dhandapani Yogeshwaran, Joseph E. Yukich. Limit theory for geometric statistics of point processes having fast decay of correlations. Annals of Probability, Institute of Mathematical Statistics, 2019, 47 (2), pp.835-895. ⟨10.1214/18-AOP1273⟩. ⟨hal-01331939⟩

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