Empirical Processes of Multidimensional Systems with Multiple Mixing Properties

TL;DR

This paper proves a multivariate empirical process central limit theorem for stationary multidimensional systems with weak dependence conditions, applicable to ergodic torus automorphisms, Markov chains, and dynamical systems with spectral gaps.

Contribution

It extends univariate empirical process CLT techniques to multivariate systems with multiple mixing properties, providing new moment bounds and broad applicability.

Findings

Established multivariate empirical process CLT under weak dependence.

Proved CLT for ergodic torus automorphisms and systems with spectral gaps.

Developed a multivariate extension of existing univariate techniques.

Abstract

We establish a multivariate empirical process central limit theorem for stationary $\R^d$-valued stochastic processes $(X_i)_{i\geq 1}$ under very weak conditions concerning the dependence structure of the process. As an application we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling, Durieu and Voln\'y \cite{DehDurVol09} in the univariate case. As an important technical ingredient, we prove a $(2p)$th moment bound for partial sums in multiple mixing systems.