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

1984/12/01 by Manfred Denker, Walter Philipp · 3 citations
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #Mathematical Dynamics and Fractals

paper · pdf · doi:10.1017/s0143385700002637

openalex publication_date 1984/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

Abstract Let 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. 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 . 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.

Citations

• Almost sure invariance principles for partial sums of weakly dependent random variables

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