A functional central limit theorem for integrals of stationary mixing random fields

TL;DR

This paper establishes a functional central limit theorem for integrals of stationary mixing random fields, describing the asymptotic behavior of these integrals as the domain expands, with implications for understanding their Gaussian limits.

Contribution

It introduces a new functional CLT for integrals of stationary mixing random fields indexed by functions, expanding the theoretical understanding of their asymptotic distributions.

Findings

Proves a functional CLT for integrals of stationary mixing random fields.

Characterizes the covariance structure of the limiting Gaussian process.

Provides conditions under which the theorem holds.

Abstract

We prove a functional central limit theorem for integrals $\int_W f(X(t))\, dt$, where $(X(t))_{t\in\mathbb{R}^d}$ is a stationary mixing random field and the stochastic process is indexed by the function $f$, as the integration domain $W$ grows in Van Hove-sense. We discuss properties of the covariance function of the asymptotic Gaussian process.