Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time

TL;DR

This paper analyzes the long-term behavior of perturbed semi-Markov processes with absorbing states, providing exponential asymptotic expansions and recursive algorithms for joint probabilities as time grows and perturbations diminish.

Contribution

It introduces a method for deriving exponential asymptotics of joint probabilities in perturbed semi-Markov processes with absorbing states, including a recursive algorithm for coefficient computation.

Findings

Derived exponential asymptotic expansions for joint probabilities.

Developed a recursive algorithm for computing expansion coefficients.

Analyzed asymptotic behavior as time tends to infinity and perturbation tends to zero.

Abstract

We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behaviour of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with an recursive algorithm for computing the coefficients in the expansions.