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Stephen Chestnut ; Manuel E. Lladser - Occupancy distributions in Markov chains via Doeblin's ergodicity coefficient

dmtcs:2789 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10) - https://doi.org/10.46298/dmtcs.2789
Occupancy distributions in Markov chains via Doeblin's ergodicity coefficientConference paper

Authors: Stephen Chestnut 1; Manuel E. Lladser 1

  • 1 Department of Applied Mathematics [Boulder]

We state and prove new properties about Doeblin's ergodicity coefficient for finite Markov chains. We show that this coefficient satisfies a sub-multiplicative type inequality (analogous to the Markov-Dobrushin's ergodicity coefficient), and provide a novel but elementary proof of Doeblin's characterization of weak-ergodicity for non-homogeneous chains. Using Doeblin's coefficient, we illustrate how to approximate a homogeneous but possibly non-stationary Markov chain of duration n by independent and short-lived realizations of an auxiliary chain of duration of order ln(n). This leads to approximations of occupancy distributions in homogeneous chains, which may be particularly useful when exact calculations via one-step methods or transfer matrices are impractical, and when asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques.


Volume: DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: compound Poisson approximation,Doeblin,Markov chain embedding technique,motif,occupancy distribution,pattern,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS],[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM],[INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG]
Funding:
    Source : OpenAIRE Graph
  • AMC-SS: Markovian Embeddings for the Analysis and Computation of Patterns in non-Markovian Random Sequences; Funder: National Science Foundation; Code: 0805950

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