Asymptotics for Quasi-Stationary Distributions of Perturbed Discrete Time Semi-Markov Processes

TL;DR

This paper develops asymptotic power series expansions for quasi-stationary distributions in perturbed discrete-time semi-Markov processes, providing a recursive algorithm for computation and illustrating it with a numerical example.

Contribution

It introduces a method to derive asymptotic expansions for quasi-stationary distributions in perturbed semi-Markov processes, including a recursive computational algorithm.

Findings

Derived asymptotic power series expansions for distributions.

Provided a recursive algorithm for coefficient computation.

Demonstrated the method with a numerical example.

Abstract

In this paper, we study quasi-stationary distributions of nonlinearly perturbed semi-Markov processes in discrete time. This type of distributions is of interest for the analysis of stochastic systems which have finite lifetimes, but are expected to persist for a long time. We obtain asymptotic power series expansions for quasi-stationary distributions and it is shown how the coefficients in these expansions can be computed from a recursive algorithm. As an illustration of this algorithm, we present a numerical example for a discrete time Markov chain.