Convergence parameters of nonnegative block tri-diagonal matrices and their application to multi-dimensional QBD processes

TL;DR

This paper extends matrix analytic methods to infinite-dimensional nonnegative block tri-diagonal matrices and applies these results to multi-dimensional QBD processes to derive bounds on stationary distribution decay rates.

Contribution

It provides new methods for analyzing convergence parameters of infinite-dimensional matrices and applies them to multi-dimensional QBD processes for the first time.

Findings

Extended matrix analytic methods to countably infinite dimensions.

Derived lower bounds for decay rates of stationary distributions.

Applied theoretical results to multi-dimensional QBD processes.

Abstract

First, we consider a nonnegative homogeneous block tri-diagonal matrix and obtain its convergence parameter, where some results in the field of matrix analytic method are extended to the case where block matrices have countably infinite dimension. Second, we apply our results to a multi-dimensional QBD process and obtain lower bounds for the directional asymptotic decay rates of the stationary distribution.