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The Central Limit Theorem for uniformly strong mixing measures

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Nicolai T A Haydn

Dynamical Systems · Submitted 7 Mar 2009

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite.

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arXiv ID0903.1325 (version 1)
Pith ID2KZW4WR5, Pith's permanent record of this paper and its review
CitationNicolai T A Haydn. "The Central Limit Theorem for uniformly strong mixing measures." arXiv:0903.1325 (2009). https://pith.science/paper/2KZW4WR5
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The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.

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s0 The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). background 1.00 0.03 0.03 asserted 1.00
s1 In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. result 0.80 0.96 0.86 asserted 1.00
s2 Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. contribution 0.74 0.69 0.08 asserted 1.00
s3 We also obtain the law of the iterated logarithm and the weak invariance principle for the information function. result 0.95 0.94 0.10 asserted 1.00

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Original abstract

The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.

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  • There is also a large body of work (see [ref] and references therein) on variants of the CLT for the empirical entropy of some ergodic mappings in the language of ergodic theory, among which, of great relevance to this work are [ref] , where CLTs with Berry-Esseen bounds are derived.

    Limit Theorems in Hidden Markov Models · Main Results and Related Work · 0.85

  • There are also other related work in different contexts from outside the information theory community, such as [ref] .

    A Randomized Approach to the Capacity of Finite-State Channels · Introduction · 0.93

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