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Stick-Breaking processes, Clumping, and Markov Chain Occupation Laws

Author

Listed:
  • Zach Dietz

    (University of Arizona)

  • William Lippitt

    (University of Arizona)

  • Sunder Sethuraman

    (University of Arizona)

Abstract

We connect the empirical or ‘occupation’ laws of certain discrete space time-inhomogeneous Markov chains, related to simulated annealing, to a novel class of ‘stick-breaking’ processes, a ‘nonexchangeable’ generalization of the Dirichlet process used in nonparametric Bayesian statistics. To make this unexpected correspondence, we examine an intermediate ‘clumped’ structure in both the time-inhomogeneous Markov chains and the stick-breaking processes, perhaps of its own interest, which records the sequence of different states visited and the scaled proportions of time spent on them. By matching the associated intermediate structures, we identify the limits of the empirical measures of the time-inhomogeneous Markov chains as types of stick-breaking processes.

Suggested Citation

  • Zach Dietz & William Lippitt & Sunder Sethuraman, 2023. "Stick-Breaking processes, Clumping, and Markov Chain Occupation Laws," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 129-171, February.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-020-00236-x
    DOI: 10.1007/s13171-020-00236-x
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    References listed on IDEAS

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    1. Günter Last, 2020. "An Integral Characterization of the Dirichlet Process," Journal of Theoretical Probability, Springer, vol. 33(2), pages 918-930, June.
    2. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265.
    3. Gantert, Nina, 1990. "Laws of large numbers for the annealing algorithm," Stochastic Processes and their Applications, Elsevier, vol. 35(2), pages 309-313, August.
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