Fluctuations of linear statistics for Gaussian perturbations of the lattice $\mathbb{Z}^d$

TL;DR

This paper investigates the asymptotic fluctuations of linear statistics for a Gaussian-perturbed lattice in olds, analyzing how these fluctuations behave as the observation window grows large, with focus on smooth and convex indicator test functions.

Contribution

It provides new insights into the fluctuation behavior of Gaussian-perturbed lattice point processes, especially for specific classes of test functions, extending understanding of their large-scale statistical properties.

Findings

Fluctuations depend on the regularity of the test function.

Asymptotic variance is characterized for different test functions.

Results apply to both stationary and non-stationary Gaussian perturbations.

Abstract

We study the point process $W$ in $\mathbb{R}^d$ obtained by adding an independent Gaussian vector to each point in $\mathbb{Z}^d$. Our main concern is the asymptotic size of fluctuations of the linear statistics in the large volume limit, defined as \[ N(h,R) = \sum_{w\in W} h\left(\frac{w}{R}\right), \] where $h\in \left(L^1\cap L^2\right)(\mathbb{R}^d)$ is a test function and $R\to \infty$. We will also consider the stationary counter-part of the process $W$, obtained by adding to all perturbations a random vector which is uniformly distributed on $[0,1]^d$ and is independent of all the Gaussians. We focus on two main examples of interest, when the test function $h$ is either smooth or is an indicator function of a convex set with a smooth boundary whose curvature does not vanish.