The Almost Sure Invariance Principle for Beta-Mixing Measures

Autor:	Nicolai Haydn
Rok vydání:	2015
Předmět:	Combinatorics Ergodic system Logarithm Invariance principle Information function Entropy (information theory) Statistical and Nonlinear Physics Almost everywhere Exponential decay Mathematical Physics Mathematics Central limit theorem
Zdroj:	Journal of Statistical Physics. 159:231-254
ISSN:	1572-9613 0022-4715
DOI:	10.1007/s10955-014-1185-6
Popis:	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 \(\fancyscript{L}^1\)-type Gibbs condition. We also prove the ASIP for the recurrence time.
Databáze:	OpenAIRE
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