Generalized inverses and their application to applied probability problems

The main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a Markov chain is present either directly or indirectly through some form of imbedding. By characterizing all generalized inverses of I—P, where P is the transition matrix of a finite irreducible discrete time Markov chain, we are able to obtain general procedures for finding stationary distributions, moments of the first passage time distributions, and asymptotic forms for the moments of the occupation-time random variables. It is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework. More generally, in the context of a Markov renewal process setting the aforementioned problems are also examined using generalized inverses of I—P. As a special case, Markov chains in continuous time are considered, and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process, instead of to I—P, where P is the transition matrix of the discrete time jump Markov chain.