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Growth and recurrence of stationary random walks
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  • Published: February 2003

Growth and recurrence of stationary random walks

  • Gernot Greschonig1 &
  • Klaus Schmidt2 

Probability Theory and Related Fields volume 125, pages 266–270 (2003)Cite this article

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Abstract

 Let (X n ,n≥1) be a real-valued ergodic stationary stochastic process, and let (Y n =X 1 +…+X n ,n≥1) be the associated random walk. We prove the following: if the sequence of distributions of the random variables Y n /n,n≥1, is uniformly tight (or, more generally, does not have the zero measure as a vague limit point), then there exists a real number c such that the random walk (Y n −nc,n≥1) is recurrent. If this sequence of distributions converges to a probability measure ρ on ℝ (or, more generally, has a nonzero limit ρ in the vague topology), then (Y n −nc,n≥1) is recurrent for ρ−a.e.cℝ.

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Authors and Affiliations

  1. Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria. e-mail: gernot.greschonig@univie.ac.at, , , , , , AT

    Gernot Greschonig

  2. Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria and Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria. e-mail: klaus.schmidt@univie.ac.at, , , , , , AT

    Klaus Schmidt

Authors
  1. Gernot Greschonig
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  2. Klaus Schmidt
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Additional information

Received: 24 September 2001 / Revised version: 1 August 2002 / Published online: 24 October 2002

The first author was partially supported by the FWF research project P14379-MAT.

Mathematics Subject Classification (2000): 37A20, 37A50, 60G10, 60G50

Key words or phrases: Recurrent stationary random walks – Recurrent cocycles

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Greschonig, G., Schmidt, K. Growth and recurrence of stationary random walks. Probab Theory Relat Fields 125, 266–270 (2003). https://doi.org/10.1007/s004400200237

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  • Issue Date: February 2003

  • DOI: https://doi.org/10.1007/s004400200237

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Keywords

  • Real Number
  • Stochastic Process
  • Probability Measure
  • Random Walk
  • Limit Point
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