Approximating class approach for empirical processes of dependent sequences indexed by functions

TL;DR

This paper investigates the weak convergence of empirical processes for dependent data, especially from dynamical systems and Markov chains, introducing a new bracketing number to handle complex index classes.

Contribution

It introduces a novel bracketing number for empirical processes indexed by classes different from the observables, applicable to dependent data with spectral gap properties.

Findings

Applicable to dynamical systems and Markov chains with spectral gap.

Extends empirical process theory to dependent sequences beyond classical cases.

Provides tools for analyzing empirical processes with complex index classes.

Abstract

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class ${\mathcal{F}}$ which fits this setting. Our results apply to the empirical process of data $(X_i)_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.