Stationary random measures : Covariance asymptotics, variance bounds and central limit theorems

TL;DR

This paper develops a comprehensive theoretical framework for analyzing covariance, variance, and central limit theorems of linear statistics of stationary random measures, with explicit formulas and asymptotic behaviors.

Contribution

It provides exact series-expansion formulas for covariance of smooth statistics, describes variance asymptotics, and proves a central limit theorem for certain point processes.

Findings

Exact covariance series-expansion formulas derived.

Variance asymptotics show even power reduction in smooth cases.

Central limit theorem established for specific point processes.

Abstract

We consider covariance asymptotics for linear statistics of general stationary random measures in terms of their truncated pair correlation measure. We give exact infinite series-expansion formulas for covariance of smooth statistics of random measures involving higher-order integrals of the truncated correlation measures and higher-order derivatives of the test functions and also equivalently in terms of their Fourier transforms. Exploiting this, we describe possible covariance and variance asymptotics for Sobolev and indicator statistics. In the smooth case, we show that that order of variance asymptotics drops by even powers and give a simple example of random measure exhibiting such a variance reduction. In the case of indicator statistics of C1-smooth sets, we derive covariance asymptotics at surface-order scale with the limiting constant depending on intersection of the boundaries of the two sets. We complement this by a lower bound for random measures with a non-trivial atomic part. Restricting to simple point processes, we prove a central limit theorem for Sobolev and Holder continuous statistics of simple point processes satifying a certain integral identity for higher-order truncated correlation functions.