Decoupling Inequalities for Stationary Gaussian Processes

Abstract

Let $\{X_n\}_{n \in \mathbb{Z}^d}$ be a stationary Gaussian process. It is proved that for all finite subsets $J$ of $\mathbb{Z}^d$ and complex-valued measurable functions $f_j, j \in J$, of a real variable, $|E(\prod_{j \in J}f_j(X_j))| \leq \prod_{j \in J}\|f_j(X_0)\|_p,$ where $p = \sum_{n \in \mathbb{Z}^d} \lbrack|E(X_0X_n)|/E(X^2_0)\rbrack$ is independent of $J$. A continuous version of this inequality is proved for stationary Gaussian processes $\{X_t\}_{t \in_\mathbb{R}^d}$. It is shown that for all bounded measurable subsets $\Lambda$ of $\mathbb{R}^d$ and complex-valued measurable functions $V$ of a real variable, $|E\big(\exp\big(\int_\Lambda V(X_t) d\mathbf{t}\big)\big)| \leq \|\exp(V(X_0))\|^{\|\Lambda\|}_p,$ where $|\Lambda|$ is the Lebesgue measure of $\Lambda$ and $p = \int_{\mathbb{R}^d} \lbrack|E(X_0X_t)|/E(X^2_0)\rbrack d\mathbf{t}$. Similar inequalities are proved for stationary Gaussian processes indexed by periodic quotient groups of $\mathbb{Z}^d$ and $\mathbb{R}^d$.