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Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings

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  • James Allen Fill

    (The Johns Hopkins University)

  • Vince Lyzinski

    (The Johns Hopkins University)

Abstract

We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hitting-time distributions for finite-state Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hitting-time distributions and the interlacing eigenvalues theorem for bordered symmetric matrices.

Suggested Citation

  • James Allen Fill & Vince Lyzinski, 2014. "Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings," Journal of Theoretical Probability, Springer, vol. 27(3), pages 954-981, September.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:3:d:10.1007_s10959-012-0457-9
    DOI: 10.1007/s10959-012-0457-9
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    References listed on IDEAS

    as
    1. Persi Diaconis & Laurent Miclo, 2009. "On Times to Quasi-stationarity for Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 22(3), pages 558-586, September.
    2. James Allen Fill, 2009. "On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains," Journal of Theoretical Probability, Springer, vol. 22(3), pages 587-600, September.
    3. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.
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