## The Annals of Probability

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

#### Abstract

Let $\mathcal{P}$ be a simple, stationary point process on $\mathbb{R}^{d}$ having fast decay of correlations, that is, its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let $\mathcal{P}_{n}:=\mathcal{P}\cap W_{n}$ be its restriction to windows $W_{n}:=[-{\frac{1}{2}}n^{1/d},{\frac{1}{2}}n^{1/d}]^{d}\subset\mathbb{R}^{d}$. 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 re-scaled, possibly signed, $\xi$-weighted point 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 nonlinear 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 neighbors graph) of $\alpha$-determinantal point processes (for $-1/\alpha\in\mathbb{N}$) having fast decreasing kernels, including the $\beta$-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [Ann. Probab. 30 (2002) 171–187] to nonlinear statistics. It also gives the limit theory for geometric $U$-statistics of $\alpha$-permanental point processes (for $1/\alpha\in\mathbb{N}$) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [Comm. Math. Phys. 310 (2012) 75–98] and Shirai and Takahashi [J. Funct. Anal. 205 (2003) 414–463], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [Stochastic Process. Appl. 56 (1995) 321–335; Statist. Probab. Lett. 36 (1997) 299–306] to show the fast decay of the correlations of $\xi$-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing, and consequently yields the asymptotic normality of $\mu_{n}^{\xi}$ via an extension of the cumulant method.

#### Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 835-895.

Dates
Revised: January 2018
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.aop/1551171639

Digital Object Identifier
doi:10.1214/18-AOP1273

Mathematical Reviews number (MathSciNet)
MR3916936

Zentralblatt MATH identifier
07053558

#### Citation

Błaszczyszyn, B.; Yogeshwaran, D.; Yukich, J. E. Limit theory for geometric statistics of point processes having fast decay of correlations. Ann. Probab. 47 (2019), no. 2, 835--895. doi:10.1214/18-AOP1273. https://projecteuclid.org/euclid.aop/1551171639

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#### Supplemental materials

• Supplement to “Limit theory for geometric statistics of point processes having fast decay of correlations”. This supplement contains various auxiliary facts needed in the proofs. These facts, some of which are of independent interest, may also be found in the arXiv version [10] of this paper.