Decoupling Inequalities for Stationary Gaussian Processes

Let {Xn}n ∈ Zd be a stationary Gaussian process. It is proved that for all finite subsets J of Zdand complex-valued measurable functions fj, j ∈ 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 = ∑n ∈ Zd [|E(X0X n)|/E(X2 0)] is independent of J. A continuous version of this inequality is proved for stationary Gaussian processes {Xt}t ∈R d . It is shown that for all bounded measurable subsets Λ of Rdand complex-valued measurable functions V of a real variable, |E(exp(∫ΛV(Xt) dt))| ≤ |exp(V(X0))||Λ| p, where |Λ| is the Lebesgue measure of Λ and p = ∫Rd [|E(X0X t)|/E(X2 0)] dt. Similar inequalities are proved for stationary Gaussian processes indexed by periodic quotient groups of Zdand Rd.