Variations around Eagleson's Theorem on mixing limit theorems for dynamical systems

TL;DR

This paper extends Eagleson's Theorem to almost sure limit theorems and mixing systems, showing that convergence in distribution implies convergence under broader conditions, with new results on conditioning at multiple times.

Contribution

It introduces a version of Eagleson's Theorem for almost sure limit theorems and mixing systems with dual-time conditioning, expanding its applicability.

Findings

Extended Eagleson's Theorem to almost sure limit theorems.

Proved results for mixing systems with conditioning at times 0 and n.

Demonstrated broader convergence conditions in dynamical systems.

Abstract

Eagleson's Theorem asserts that, given a probability-preserving map, ifrenormalized Birkhoff sums of a function converge in distribution, thenthey also converge with respect to any probability measure which isabsolutely continuous with respect to the invariant one. We prove a versionof this result for almost sure limit theorems, extending results ofKorepanov. We also prove a version of this result, in mixing systems, whenone imposes a conditioning both at time 0 and at time n.