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 -