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Random Markov processes for countable and uncountable alphabets

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Abstract

In this paper, we explore two notions of stationary processes — uniform martingales and random Markov processes. We show that every stationary process on a countable alphabet that is a uniform martingale and is dominated by a finite measure on that alphabet is also a random Markov process, and that the random look-back times and associated coupling can be chosen so that the distribution on the present given both the n-past and a look-back time of n is ‘deterministic’: all probabilities are in {0, 1}. In the case of finite alphabets, those random Markov processes for which the look-back time can be chosen with finite expected value are characterized. For stationary processes on an uncountable alphabet, a stronger condition is also considered which is sufficient to imply that a process is a random Markov processes. In addition, a number of examples are given throughout to show the sharpness of the results.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ, 85287-1804, USA

    Neal Bushaw

  2. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brasil

    Neal Bushaw

  3. Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada

    Karen Gunderson

  4. Department of Mathematics, University of Memphis, 3725 Norriswood, Memphis, TN, 38152, USA

    Steven Kalikow

Authors
  1. Neal Bushaw
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  2. Karen Gunderson
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  3. Steven Kalikow
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Corresponding author

Correspondence to Neal Bushaw.

Additional information

The second author was supported in part by the Heilbronn Institute for Mathematical Research.

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Bushaw, N., Gunderson, K. & Kalikow, S. Random Markov processes for countable and uncountable alphabets. Isr. J. Math. 216, 181–214 (2016). https://doi.org/10.1007/s11856-016-1407-5

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  • DOI: https://doi.org/10.1007/s11856-016-1407-5

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