Asymptotics of the occupation measure defined on a nonnegative matrix of two-dimensional quasi-birth-and-death type

TL;DR

This paper analyzes the asymptotic behavior of occupation and hitting measures in a two-dimensional quasi-birth-and-death process, revealing decay rates and duality relationships for these measures.

Contribution

It provides new asymptotic results for occupation and hitting measures in 2d-QBD processes, highlighting their decay rates and duality properties.

Findings

Asymptotic decay rates of occupation measure in arbitrary directions.

Duality relationship between occupation and hitting measures.

Asymptotic properties of the hitting measure derived from occupation measure results.

Abstract

We consider a nonnegative matrix having the same block structure as that of the transition probability matrix of a two-dimensional quasi-birth-and-death process (2d-QBD process for short) and define two kinds of measure for the nonnegative matrix. One corresponds to the mean number of visits to each state before the 2d-QBD process starting from the level zero returns to the level zero for the first time. The other corresponds to the probabilities that the 2d-QBD process starting from each state visits the level zero. We call the former the occupation measure and the latter the hitting measure. We obtain asymptotic properties of the occupation measure such as the asymptotic decay rate in an arbitrary direction. Those of the hitting measure can be obtained from the results for the occupation measure by using a kind of duality between the two measures.