Quasi-stationary distributions in reducible state spaces

TL;DR

This paper analyzes quasi-stationary distributions in reducible Markov chains, providing explicit characterizations of convergence speeds, polynomial correction factors, and conditions for existence without irreducibility.

Contribution

It introduces assumptions for reducible state spaces, characterizes polynomial convergence parameters, and extends results to non-irreducible chains with explicit convergence estimates.

Findings

Explicit characterization of polynomial convergence parameters

Complete set of quasi-stationary distributions identified

Proven existence of quasi-stationary distributions without irreducibility

Abstract

We study quasi-stationary distributions and quasi-limiting behavior of Markov chains in general reducible state spaces with absorption. We propose a set of assumptions dealing with particular situations where the state space can be decomposed into three subsets between which communication is only possible in a single direction. These assumptions allow us to characterize the exponential order of magnitude and the exact polynomial correction, called polynomial convergence parameter, for the leading order term of the semigroup for large time. They also provide explicit convergence speeds to this leading order term. We apply these results to general Markov chains with finitely or denumerably many communication classes using a specific induction over the communication classes of the chain. We are able to explicitely characterize the polynomial convergence parameter, to determine the complete set of quasistationary distributions and to provide explicit estimates for the speed of convergence to quasi-limiting distributions in the case of finitely many communication classes. We conclude with an application of these results to the case of denumerable state spaces, where we are able to prove that, in general, there is existence of a quasi-stationary distribution without assuming irreducibility before absorption. This actually holds true assuming only aperiodicity, the existence of a Lyapunov function and the existence of a point in the state space from which the return time is finite with positive probability.