A central limit theorem for stationary random fields

TL;DR

This paper proves a central limit theorem and invariance principle for a broad class of stationary random fields, including spatial processes, under simple dependence conditions, with applications to sample auto-covariance functions.

Contribution

It establishes a general CLT and invariance principle for stationary random fields of the form g(ε), with minimal assumptions on the domain.

Findings

CLT holds for stationary random fields under short-range dependence.

Invariance principle is established for these fields.

Limit theorem for sample auto-covariance function is proven.

Abstract

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k = g(\varepsilon_{k-s}, s \in \Z^d)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d random variables and $g$ is a measurable function. Such kind of spatial processes provides a general framework for stationary ergodic random fields. Under a short-range dependence condition, we show that the central limit theorem holds without any assumption on the underlying domain on which the process is observed. A limit theorem for the sample auto-covariance function is also established.