A central limit theorem for stationary random fields

Mohamed El Machkouri (1) , Dalibor Volny (1) , Wei Biao Wu (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Department of Mathematics

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 $X_k = g\left(\varepsilon_{k-s}, s \in \Z^d \right)$, $k\in\Z^d$, 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.

Mots clés: Central limit theorem; spatial processes; m-dependent random fields; weak mixing.; weak mixing

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

Mohamed El Machkouri, Dalibor Volny, Wei Biao Wu. A central limit theorem for stationary random fields. 2012. ⟨hal-00619049v2⟩