Hitting times and interlacing eigenvalues: a stochastic approach using intertwinings

TL;DR

This paper introduces a matrix-analytic method based on intertwinings of Markov semigroups to analyze hitting-time distributions in finite-state Markov chains, providing new proofs and stochastic constructions that deepen understanding.

Contribution

It develops a systematic approach using intertwinings to analyze hitting times, offering new proofs and insights, including a novel connection to interlacing eigenvalues.

Findings

New proofs for Brown's theorems on hitting-time distributions

Stochastic constructions that clarify hitting-time behaviors

Connection between hitting times and interlacing eigenvalues

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.