  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''.
