# Skip-free Markov chains

## TL;DR

This paper develops a comprehensive theory for skip-free Markov chains, including potential and fluctuation theories, spectral properties, and convergence behavior, expanding understanding of their structure and dynamics.

## Contribution

It introduces a general framework for skip-free Markov chains, characterizes their potential kernel, fluctuation identities, spectral properties, and conditions for cutoff phenomena.

## Key findings

- Explicit potential kernel characterization
- Generalized fluctuation identities
- Conditions for spectral gap and cutoff

## 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 family of fundamental excessive functions, which are defined by means of the theory of 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 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.