An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

TL;DR

This paper establishes a general empirical process central limit theorem for multidimensional dependent data, especially useful for dynamical systems and Markov chain functionals, extending previous results with broader applicability.

Contribution

It introduces new, general conditions for weak convergence of empirical processes involving dependent data, improving upon earlier methods and enabling new applications.

Findings

Provides a set of conditions for weak convergence of empirical processes

Extends previous results to broader classes of dependent data

Enables applications to dynamical systems and Markov chain functionals

Abstract

Let $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications.