Light-Tailed Asymptotics of Stationary Tail Probability Vectors of Markov Chains of M/G/1 Type

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.

Keywords:

• Dominant pole
• Generating function
• Light-tailed asymptotics
• Markov chain of M/G/1 type
• Stationary tail probability vector

Mathematics Subject Classification:

• Primary 60K25
• Secondary 60J10

ACKNOWLEDGMENTS

The authors thank to Takine Tetsuya and Masakiyo Miyazawa for his valuable comments on an early version of this paper. They also thank the anonymous referees and the editor for their suggestions on how to strengthen the results and improve the presentation of this paper. Research of the third author was supported in part by Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science under Grant No. 21710151.