Hitting times and interlacing eigenvalues: a stochastic approach using intertwinings

2012/01/31 by James Allen Fill, Fill, James Allen, Vince Lyzinski +1
Mathematics · #60J10 (Primary) 15A42 (Secondary) #FOS: Mathematics #Probability (math.PR) #math.PR #msc:15A42 #msc:60J10

paper · pdf · doi:10.48550/arxiv.1201.6441

arxiv created 2012/09/01 · arxiv updated 2012/09/04

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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