Skip-free Markov chains



Choi, Michael CH and Patie, Pierre ORCID: 0000-0003-4221-0439
(2019) Skip-free Markov chains. Transactions of the American Mathematical Society, 371 (10). 7301 - 7342.

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Abstract

The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of the family of fundamental excessive functions, which are defined by means of the theory of the Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to identify the class of skip-free Markov chains whose transition operator has only real and simple eigenvalues. We manage to find a set of sufficient and easy-to-check conditions on the one-step transition probability for a Markov chain to belong to this class. We also study several properties of this class including their spectral expansions given in terms of a Riesz basis, derive a necessary and sufficient condition for this class to exhibit a separation cutoff, and give a tighter bound on its convergence rate to stationarity than existing results.

Item Type: Article
Uncontrolled Keywords: Markov chains, potential theory, Martin boundary, fluctuation theory, spectral theory, non-self=adjoint operator, rate of convergence, cutoff
Depositing User: Symplectic Admin
Date Deposited: 27 Apr 2020 10:18
Last Modified: 06 Jan 2022 08:29
DOI: 10.1090/tran/7773
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3081727