Empirical process sampled along a stationary process

Let $(X_{\underset{‾}{\mathrm{\ell }}}{)}_{\underset{‾}{\mathrm{\ell }}\in {\mathbb{Z}}^d}$ be a real random field (r.f.) indexed by ${\mathbb{Z}}^d$ with common probability distribution function $F$. Let $(z_k{)}_{k=0}^{\mathrm{\infty }}$ be a sequence in ${\mathbb{Z}}^d$. The empirical process obtained by sampling the random field along $(z_k)$ is ${\sum }_{k=0}^{n-1}[{\mathbf{1}}_{X_{z_k}\le s}-F(s)]$. 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 $X_{\underset{‾}{\mathrm{\ell }}}$‘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’'.