Empirical process sampled along a stationary process

Abstract

Let $(X_{\underline{\ell}})_{\underline{\ell} \in \mathbb Z^d}$ be a real random field (r.f.) indexed by $\mathbb Z^d$ with common probability distribution function $F$. Let $(z_k)_{k=0}^\infty$ be a sequence in $\mathbb Z^d$. The empirical process obtained by sampling the random field along $(z_k)$ is $\sum_{k=0}^{n-1} [{\bf 1}_{X_{z_k} \leq s}- F(s)]$. We give conditions on $(z_k)$ implying the Glivenko-Cantelli theorem for the empirical process sampled along $(z_k)$ in different cases (independent, associated or weakly correlated random variables). We consider also the functional central limit theorem when the $X_{\underline{\ell}}$'s are i.i.d. These conditions are examined when $(z_k)$ is provided by an auxiliary stationary process in the framework of ``random ergodic theorems''.