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An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables
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  • Published: June 1990

An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables

  • Erich Berger1 nAff2 

Probability Theory and Related Fields volume 84, pages 161–201 (1990)Cite this article

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Summary

In this paper we establish an almost sure invariance principle with an error termo((t log logt)1/2) (ast→∞) for partial sums of stationary ergodic martingale difference sequences taking values in a real separable Banach space. As partial sums of weakly dependent random variables can often be well approximated by martingales, this result also leads to almost sure invariance principles for a wide class of stationary ergodic sequences such as ø-mixing and α-mixing sequences and functionals of such sequences. Compared with previous related work for vector valued random variables (starting with an article by Kuelbs and Philipp [27]), the present approach leads to a unification of the theory (at least for stationary sequences), moment conditions required by earlier authors are relaxed (only second order weak moments are needed), and our proofs are easier in that we do not employ estimates of the rate of convergence in the central limit theorem but merely the central limit theorem itself.

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Author notes
  1. Erich Berger

    Present address: Fachbereich Mathematik WE 3, Freie Universität Berlin, Arnimallee 2-6, D-1000, Berlin 33

Authors and Affiliations

  1. Institut für Mathematische Stochastik, Universität Göttingen, Lotzestrasse 13, D-3400, Göttingen, Germany

    Erich Berger

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  1. Erich Berger
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Berger, E. An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables. Probab. Th. Rel. Fields 84, 161–201 (1990). https://doi.org/10.1007/BF01197844

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  • Received: 09 September 1988

  • Revised: 06 April 1989

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01197844

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Keywords

  • Banach Space
  • Central Limit Theorem
  • Moment Condition
  • Stationary Sequence
  • Invariance Principle
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