Subexponential Asymptotics of the Stationary Distributions of GI/G/1-Type Markov Chains

Abstract

This article considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For case (i), we present a weaker sufficient condition for the subexponential asymptotics than those given in the literature. As for case (ii), the subexponential asymptotics has not been studied, as far as we know. We show that the subexponential asymptotics in case (ii) is different from that in case (i). We also study the locally subexponential asymptotics of the stationary distributions in both cases (i) and (ii).

Keywords:

• GI/G/1-type Markov chain
• (Locally) subexponential
• Markov additive process (MAdP)
• Stationary distribution

Mathematics Subject Classification:

• Primary 60K25
• Secondary 60J10

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Corrigendum

ACKNOWLEDGMENT

The authors thank Professors Bara Kim and Jeongsim Kim for providing a copy of their manuscript prior to publication. Research of the second author was supported in part by Grant-in-Aid for Young Scientists (B) of the Japan Society for the Promotion of Science under Grant No. 24710165.