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  • Ergodic Theory and Dynamical Systems, Volume 4, Issue 4
  • December 1984, pp. 541-552

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

  • Manfred Denker (a1) and Walter Philipp (a2)
  • DOI:
  • Published online: 01 September 2008

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.

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[4]I. A. Ibragimov . Some limit theorems for stationary processes. Theor. Probability Appl. 7 (1962) 349382.

[6]M. Ratner . The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973), 181197.

[8]D. Ruelle , Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9 (1968), 267278.

[9]R. J. Serfling , Moment inequalities for the maximum cumulative sum. Ann. Math. Stat. 41 (1970), 12271234

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