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A strong invariance principle for nonconventional sums
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  • Published: 08 December 2011

A strong invariance principle for nonconventional sums

  • Yuri Kifer1 

Probability Theory and Related Fields volume 155, pages 463–486 (2013)Cite this article

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An Erratum to this article was published on 21 February 2013

Abstract

In Kifer and Varadhan (Nonconventional limit theorems in discrete and continuous time via martingales, 2010) we obtained a functional central limit theorem (known also as a weak invariance principle) for sums of the form \({\sum_{n=1}^{[Nt]} F\big(X(n), X(2n), .\, .\, .\, .\, X(kn), X(q_{k+1}(n)), X(q_{k+2}(n)), .\, .\, .\, , X(q_\ell(n))\big)}\) (normalized by \({1/\sqrt N}\)) where X(n), n ≥ 0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties and q i , i > k are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q i ’s are polynomials of growing degrees. This paper deals with strong invariance principles (known also as strong approximation theorems) for such sums which provide their uniform in time almost sure approximation by processes built out of Brownian motions with error terms growing slower than \({\sqrt N}\) . This yields, in particular, an invariance principle in the law of iterated algorithm for the above sums. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems as well, as to some questions in metric number theory and they can be considered as a natural follow up of a series of papers dealing with nonconventional ergodic averages.

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Authors and Affiliations

  1. Institute of Mathematics, Hebrew University, 91904, Jerusalem, Israel

    Yuri Kifer

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  1. Yuri Kifer
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Correspondence to Yuri Kifer.

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Cite this article

Kifer, Y. A strong invariance principle for nonconventional sums. Probab. Theory Relat. Fields 155, 463–486 (2013). https://doi.org/10.1007/s00440-011-0404-1

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  • Received: 23 December 2010

  • Revised: 25 October 2011

  • Published: 08 December 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0404-1

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Keywords

  • Strong approximations
  • Limit theorems
  • Martingale approximation
  • Mixing
  • Dynamical systems

Mathematics Subject Classification (2000)

  • Primary 60F15
  • Secondary 60G42
  • 37D20
  • 60F17
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