# Rate and G- matrices of a non-negative block tri-diagonal matrix and application to a 3-dimensional skip-free Markov modulated random walk

Research paper by Toshihisa Ozawa

Indexed on: 08 Nov '16Published on: 08 Nov '16Published in: arXiv - Mathematics - Probability

#### Abstract

First, we extend matrix analytic methods to the case of nonnegative block tri-diagonal matrices with countably many phase states, where rate matrices and so-called G-matrices are redefined and some properties of them are clarified. Second, we apply the results to a three-dimensional skip-free Markov modulated random walk (3D-MMRW for short) and obtain a lower bound for the directional asymptotic decay rates of each row of the fundamental matrix (occupation measure) arising from the 3D-MMRW.