Empirical central limit theorems for ergodic automorphisms of the torus

TL;DR

This paper establishes empirical central limit theorems for ergodic automorphisms of the torus, demonstrating weak convergence of empirical processes under mild regularity conditions on the functions involved.

Contribution

It introduces new limit theorems, inequalities, and estimates for non-adapted sequences in the context of ergodic automorphisms of the torus.

Findings

Proves weak convergence of empirical processes for ergodic toral automorphisms.

Develops new inequalities for non-adapted sequences.

Provides estimates of conditional expectations for functions on the torus.

Abstract

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f o T^i)_{i \geq 1} under some mild conditions on the modulus of continuity of f. The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural filtration.