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Random motions, classes of ergodic Markov chains and beta distributions

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  • Stoyanov, Jordan
  • Pirinsky, Christo

Abstract

We consider classes of discrete time Markov chains with continuous state space, the interval (0,1). These chains arise as stochastic models of phenomena in areas such as population theory, motion of particles in a random environment, etc. We exploit the Fréchet-Shohat theorem to establish that these Markov chains are ergodic and find explicitly their ergodic distributions as being beta distributions. Then we show that the convergence in total variation norm is at a geometric rate. Related topics are also discussed.

Suggested Citation

  • Stoyanov, Jordan & Pirinsky, Christo, 2000. "Random motions, classes of ergodic Markov chains and beta distributions," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 293-304, November.
    • Handle: RePEc:eee:stapro:v:50:y:2000:i:3:p:293-304
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    References listed on IDEAS

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    1. Devroye, Luc & Letac, Gerard & Seshadri, Vanamamalai, 1986. "The limit behavior of an interval splitting scheme," Statistics & Probability Letters, Elsevier, vol. 4(4), pages 183-186, June.
    2. Unknown, 1986. "Letters," Choices: The Magazine of Food, Farm, and Resource Issues, Agricultural and Applied Economics Association, vol. 1(4), pages 1-9.
    3. Robert B. Lund & Richard L. Tweedie, 1996. "Geometric Convergence Rates for Stochastically Ordered Markov Chains," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 182-194, February.
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    Cited by:

    1. Ladjimi, Fetima & Peigné, Marc, 2019. "On the asymptotic behavior of the Diaconis–Freedman chain on [0,1]," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 1-11.
    2. McKinlay, Shaun, 2017. "On beta distributed limits of iterated linear random functions," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 33-41.
    3. Pacheco-González, Carlos G., 2009. "Ergodicity of a bounded Markov chain with attractiveness towards the centre," Statistics & Probability Letters, Elsevier, vol. 79(20), pages 2177-2181, October.
    4. Letac, Gérard, 2002. "Donkey walk and Dirichlet distributions," Statistics & Probability Letters, Elsevier, vol. 57(1), pages 17-22, March.
    5. Liu, Yujie & Niu, Minwen & Yao, Dacheng & Zhang, Hanqin, 2022. "Stationary distributions and ergodicity of reflection-type Markov chains," Statistics & Probability Letters, Elsevier, vol. 189(C).

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