Empirical process sampled along a stationary process

TL;DR

This paper investigates the convergence properties of empirical processes sampled along stationary and other dependent random fields, establishing conditions for the Glivenko-Cantelli and functional central limit theorems.

Contribution

It provides new conditions under which empirical processes sampled along dependent random fields satisfy classical limit theorems, extending existing results to broader dependence structures.

Findings

Conditions for Glivenko-Cantelli theorem under dependence

Functional CLT for i.i.d. random fields

Analysis of sampling sequences from stationary processes

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''.