Hitting times and interlacing eigenvalues: a stochastic approach using intertwinings

James Allen Fill; Vince Lyzinski

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arXiv · 1201.6441

Hitting times and interlacing eigenvalues: a stochastic approach using intertwinings

James Allen Fill·Vince Lyzinski

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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.

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James Allen Fill, Vince Lyzinski. 2012-09-01. Hitting times and interlacing eigenvalues: a stochastic approach using intertwinings. https://arxiv.org/abs/1201.6441

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