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Approximation by Brownian motion for Gibbs measures and flows under a function. (English) Zbl 0554.60077
Let \(\{S_{\tau}:\) \(\tau\in {\mathbb{R}}\}\) denote a flow built under a Hölder-continuous function l over the base (\(\Sigma\),\(\mu)\) where \(\Sigma\) is a topological Markov chain and \(\mu\) some (\(\psi\)-mixing) Gibbs measure. For a certain class of functions f with finite \(2+\delta\)- moments it is shown that there exists a Brownian motion B(t) with respect to \(\mu\) and \(\sigma^ 2>0\) such that \(\mu\)-a.e. \[ \sup_{0\leq u<l(x)}| \int^{t}_{0}fS_{\tau}d\tau -B(\sigma^ 2t)| \ll t^{-\lambda}\quad for\quad some\quad 0<\lambda <\delta /588. \] One can also approximate in the same way by a Brownian motion \(B^*(t)\) with respect to the probability \((\int l du)^{-1}l d\mu\). 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 M. Ratner, Isr. J. Math. 16(1973), 181-197 (1974; Zbl 0283.58010), is extended.

60J65 Brownian motion
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
Full Text: DOI
[1] DOI: 10.1137/1107036 · Zbl 0119.14204 · doi:10.1137/1107036
[2] Doob, Stochastic Processes (1953)
[3] Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms 470 (1975) · Zbl 0308.28010 · doi:10.1007/BFb0081279
[4] Billingsley, Convergence of Probability Measures (1968)
[5] Philipp, Mem. Amer. Math. Soc. 161 pp none– (1975)
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[9] DOI: 10.1007/BF02757869 · Zbl 0283.58010 · doi:10.1007/BF02757869
[10] DOI: 10.1214/aoms/1177696898 · Zbl 0272.60013 · doi:10.1214/aoms/1177696898
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