Stephen Chestnut ; Manuel E. Lladser
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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)
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https://doi.org/10.46298/dmtcs.2789
Occupancy distributions in Markov chains via Doeblin's ergodicity coefficientArticle
Authors: Stephen Chestnut 1; Manuel E. Lladser 1
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Stephen Chestnut;Manuel E. Lladser
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)
AMC-SS: Markovian Embeddings for the Analysis and Computation of Patterns in non-Markovian Random Sequences; Funder: National Science Foundation; Code: 0805950
Loïc Hervé;James Ledoux, 2021, Asymptotic of products of Markov kernels. Application to deterministic and random forward/backward products, Statistics & Probability Letters, 179, pp. 109204, 10.1016/j.spl.2021.109204, https://doi.org/10.1016/j.spl.2021.109204.