TY - JOUR

T1 - Approximation by Brownian motion for Gibbs measures and flows under a function

AU - Denker, Manfred

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1984/12

Y1 - 1984/12

N2 - 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 σ2 > 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.

AB - 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 σ2 > 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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U2 - 10.1017/S0143385700002637

DO - 10.1017/S0143385700002637

M3 - Article

AN - SCOPUS:0000454046

SN - 0143-3857

VL - 4

SP - 541

EP - 552

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 4

ER -