# 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 [67] 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 [50], 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 <>
Submitted on : Tuesday, June 14, 2016 - 7:04:33 PM
Last modification on : Thursday, November 14, 2019 - 9:17:10 AM

### Citation

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

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