Empirical Processes of Multidimensional Systems with Multiple Mixing Properties

2010/04/07 by Herold Dehling, Dehling, Herold, Olivier Durieu +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.1004.1088

openalex publication_date 2010/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a multivariate empirical process central limit theorem for stationary \R^d-valued stochastic processes (X_i)_{i≥ 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 Volný \citeDehDurVol09 in the univariate case. As an important technical ingredient, we prove a (2p)th moment bound for partial sums in multiple mixing systems.

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