The Almost Sure Invariance Principle for Beta-Mixing measures

TL;DR

This paper proves that for beta-mixing measures, the information function and recurrence times follow the almost sure invariance principle, extending previous results to a broader class of mixing measures.

Contribution

It establishes the ASIP for the information function and recurrence times under beta-mixing conditions, generalizing prior results beyond strong mixing and Gibbs measures.

Findings

The logarithm of the measure of n-cylinders satisfies the ASIP for beta-mixing measures.

Recurrence times also follow the ASIP under the same conditions.

Extends the applicability of ASIP to a wider class of ergodic systems.

Abstract

The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system of finite entropy the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In addition the measure of $n$-cylinders is in various settings known to be lognormally distributed in the limit. In this paper the logarithm of the measure of $n$-cylinder, the information function, satisfies the almost sure invariance principle in the case in which the measure is $\beta$-mixing. We get a similar result for the recurrence time. Previous results are due to Philipp and Stout who deduced the ASIP when the measure is strong mixing and satisfies an $\mathscr{L}^1$-type Gibbs condition.