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

  • Manfred Denker (a1) and Walter Philipp (a2)
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 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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References
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[1]Billingsley P.. Convergence of Probability Measures. J. Wiley: New York, 1968.
[2]Bowen R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. 470. Springer: Berlin-Heidelberg-New York, 1975.
[3]Doob J. L.. Stochastic Processes. J. Wiley: New York, 1953.
[4]Ibragimov I. A.. Some limit theorems for stationary processes. Theor. Probability Appl. 7 (1962) 349382.
[5]Philipp W. & Stout W.. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 161 (1975).
[6]Ratner M.. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973), 181197.
[7]Renyi A.. Wahrscheinlichkeitsrechnung. Deutscher Verlag Wissensch.: Berlin, 1962.
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[9]Serfling R. J., Moment inequalities for the maximum cumulative sum. Ann. Math. Stat. 41 (1970), 12271234
[10]Sinai Y. G., Gibbs measures in ergodic theory. Uspeki Mat. Nauk (4) 27 (1972), 2163.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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