Approximation by Brownian motion for Gibbs measures and flows under a function

Manfred Denker

Research output: Contribution to journalArticle

68 Citations (Scopus)

Abstract

Let [formula-omitted] denote a flow built under a Hölder-continuous function l over the base (Σ, μ) where Σ is a topological Markov chain and μ some (ψ-mining) Gibbs measure. For a certain class of functions f with finite 2 + δ-moments it is shown that there exists a Brownian motion B(t) with respect to μ and σ 2 > 0 such that μ-a.e. [formula-omitted] for some 0 < λ < 5δ/588. One can also approximate in the same way by a Brownian motion B*(t) with respect to the probability [formula-omitted]. From this, the central limit theorem, the weak invariance principle, the law of the iterated logarithm and related probabilistic results follow immediately. In particular, the result of Ratner ([6]) is extended.

Original languageEnglish (US)
Pages (from-to)541-552
Number of pages12
JournalErgodic Theory and Dynamical Systems
Volume4
Issue number4
DOIs
StatePublished - Dec 1984

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Gibbs Measure
Brownian movement
Brownian motion
Approximation
Invariance
Weak Invariance Principle
Markov processes
Law of the Iterated Logarithm
Central limit theorem
Immediately
Mining
Markov chain
Continuous Function
Denote
Moment

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "Let [formula-omitted] denote a flow built under a H{\"o}lder-continuous function l over the base (Σ, μ) where Σ is a topological Markov chain and μ some (ψ-mining) Gibbs measure. For a certain class of functions f with finite 2 + δ-moments it is shown that there exists a Brownian motion B(t) with respect to μ and σ 2 > 0 such that μ-a.e. [formula-omitted] for some 0 < λ < 5δ/588. One can also approximate in the same way by a Brownian motion B*(t) with respect to the probability [formula-omitted]. From this, the central limit theorem, the weak invariance principle, the law of the iterated logarithm and related probabilistic results follow immediately. In particular, the result of Ratner ([6]) is extended.",
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Approximation by Brownian motion for Gibbs measures and flows under a function. / Denker, Manfred.

In: Ergodic Theory and Dynamical Systems, Vol. 4, No. 4, 12.1984, p. 541-552.

Research output: Contribution to journalArticle

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