Moving Averages of Homogeneous Random Fields

Abstract

Let $X(g)$ be a homogeneous random field on a discrete locally compact Abelian group $G$. Let $H(X)$ be the linear completion of $\{X(g): g \in G\}$ in $L_2$ space. The following result is obtained: there exists a fundamental random field $Y(g)$ on $G$ with values in $H(X)$ such that $X(g)$ is obtained as a moving average of $Y(g)$ if, and only if, $X(g)$ has a spectral density which is positive almost everywhere with respect to the Haar measure on the dual group of $G$.