### Abstract

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 σ
^{2}
> 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.

Original language | English (US) |
---|---|

Pages (from-to) | 541-552 |

Number of pages | 12 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1984 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*4*(4), 541-552. https://doi.org/10.1017/S0143385700002637

}

*Ergodic Theory and Dynamical Systems*, vol. 4, no. 4, pp. 541-552. https://doi.org/10.1017/S0143385700002637

**Approximation by Brownian motion for Gibbs measures and flows under a function.** / Denker, Manfred Heinz.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximation by Brownian motion for Gibbs measures and flows under a function

AU - Denker, Manfred Heinz

PY - 1984/1/1

Y1 - 1984/1/1

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 σ 2 > 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 σ 2 > 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.

UR - http://www.scopus.com/inward/record.url?scp=0000454046&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000454046&partnerID=8YFLogxK

U2 - 10.1017/S0143385700002637

DO - 10.1017/S0143385700002637

M3 - Article

VL - 4

SP - 541

EP - 552

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -