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DOOB-MARTIN COMPACTIFICATION OF A MARKOV CHAIN FOR GROWING RANDOM WORDS SEQUENTIALLY.

Hye Soo Choi1, Steven N Evans1

  • 1Department of Statistics #3860, 367 Evans Hall, University of California, Berkeley, CA 94720-3860, USA.

Stochastic Processes and Their Applications
|October 3, 2017
PubMed
Summary
This summary is machine-generated.

This study characterizes the Doob-Martin boundary of a Markov chain generating random words. It reveals how these chains behave at large times, connecting them to random total orders and probability measures.

Keywords:
Plackett-Luce modelbinomial coefficientbridgeexchangeabilityharmonic functionshufflesub-word countingvase model

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Area of Science:

  • Probability Theory
  • Stochastic Processes
  • Combinatorics

Background:

  • Markov chains are fundamental in modeling systems with sequential states.
  • Understanding the long-term behavior of Markov chains is crucial for various applications.
  • The Doob-Martin boundary provides insights into the asymptotic behavior of Markov chains.

Purpose of the Study:

  • To characterize the Doob-Martin boundary of a specific Markov chain generating random words.
  • To delineate the conditioning of this Markov chain's behavior at large times.
  • To establish correspondences between the boundary, random total orders, and probability measures.

Main Methods:

  • Analysis of a Markov chain generating sequences of words with equal numbers of 'a' and 'b'.
  • Characterization of the Doob-Martin boundary through convergence properties of word sequences.
  • Exhibition of bijective correspondences between boundary points, random total orders, and probability measures.

Main Results:

  • A concrete characterization of the Doob-Martin boundary for the studied Markov chain is obtained.
  • Convergence criteria for word sequences to boundary points are established.
  • Bijective mappings are demonstrated between the boundary, specific random total orders, and pairs of probability measures.

Conclusions:

  • The study provides a comprehensive understanding of the asymptotic behavior of the Markov chain.
  • The findings connect combinatorial structures (random words) with probabilistic concepts (boundary, measures).
  • The established correspondences offer new perspectives on random orderings and their underlying distributions.