Light-tailed asymptotics of stationary tail probability vectors of Markov chains of M/G/1 type

TL;DR

This paper investigates the light-tailed asymptotic behavior of stationary tail probabilities in Markov chains of M/G/1 type, including both typical and atypical decay patterns, by analyzing the generating functions and dominant poles.

Contribution

It extends existing studies by analyzing atypical decay cases and deriving explicit asymptotic formulas for stationary tail probabilities in M/G/1 type Markov chains.

Findings

Derived asymptotic formulas for tail probabilities.

Identified conditions for periodic and non-typical decay.

Discussed positivity of dominant asymptotic terms.

Abstract

This paper studies the light-tailed asymptotics of the stationary tail probability vectors of a Markov chain of M/G/1 type. Almost all related studies have focused on the typical case, where the transition block matrices in the non-boundary levels have a dominant impact on the decay rate of the stationary tail probability vectors and their decay is aperiodic. In this paper, we study not only the typical case but also atypical cases such that the stationary tail probability vectors decay periodically and/or their decay rate is determined by the tail distribution of jump sizes from the boundary level. We derive light-tailed asymptotic formulae for the stationary tail probability vectors by locating the dominant poles of the generating function of the sequence of those vectors. Further we discuss the positivity of the dominant terms of the obtained asymptotic formulae.