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Mixing properties of the generalized T, T-1-process

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Journal d’Analyse Mathematique Aims and scope

Abstract

Consider a general random walk on ℤd together with an i.i.d. random coloring of ℤd. TheT, T -1-process is the one where time is indexed by ℤ, and at each unit of time we see the step taken by the walk together with the color of the newly arrived at location. S. Kalikow proved that ifd = 1 and the random walk is simple, then this process is not Bernoulli. We generalize his result by proving that it is not Bernoulli ind = 2, Bernoulli but not Weak Bernoulli ind = 3 and 4, and Weak Bernoulli ind ≥ 5. These properties are related to the intersection behavior of the past and the future of simple random walk. We obtain similar results for general random walks on ℤd, leading to an almost complete classification. For example, ind = 1, if a step of sizex has probability proportional to l/|x|α (x ⊋ 0), then theT, T -1-process is not Bernoulli when α ≥2, Bernoulli but not Weak Bernoulli when 3/2 ≤α < 2, and Weak Bernoulli when 1 < α < 3/2.

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Authors and Affiliations

  1. Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525, ED Nijmegen, The Netherlands

    Frank den Hollander

  2. Department of Mathematics, Chalmers University of Technology, S-41296, Gothenburg, Sweden

    Jeffrey E. Steif

Authors
  1. Frank den Hollander
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  2. Jeffrey E. Steif
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Correspondence to Frank den Hollander.

Additional information

Research partially carried out while a guest of the Department of Mathematics, Chalmers University of Technology, Sweden in January 1996.

Research supported by grants from the Swedish Natural Science Research Council and from the Royal Swedish Academy of Sciences.

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den Hollander, F., Steif, J.E. Mixing properties of the generalized T, T-1-process. J. Anal. Math. 72, 165–202 (1997). https://doi.org/10.1007/BF02843158

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  • DOI: https://doi.org/10.1007/BF02843158

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