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

Published online by Cambridge University Press:  19 September 2008

Manfred Denker  and
Walter Philipp
Manfred Denker
Affiliation:
Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13 D-3400 Gottingen, West Germany;
Walter Philipp
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
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Abstract

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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 σ2 > 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 4 , December 1984 , pp. 541 - 552
Copyright
Copyright © Cambridge University Press 1984

References

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