Institute of Natural and Mathematical Sciences
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Item A survey of generalized inverses and their use in stochastic modelling(Massey University, 2000) Hunter, Jeffrey J.In many stochastic models, in particular Markov chains in discrete or continuous time and Markov renewal processes, a Markov chain is present either directly or indirectly through some form of embedding. The analysis of many problems of interest associated with these models, eg. stationary distributions, moments of first passage time distributions and moments of occupation time random variables, often concerns the solution of a system of linear equations involving I – P, where P is the transition matrix of a finite, irreducible, discrete time Markov chain. Generalized inverses play an important role in the solution of such singular sets of equations. In this paper we survey the application of generalized inverses to the aforementioned problems. The presentation will include results concerning the analysis of perturbed systems and the characterization of types of generalized inverses associated with Markovian kernels.Item Generalized inverses, stationary distributions and mean first passage times with applications to perturbed Markov chains(Massey University, 2002) Hunter, Jeffrey J.In an earlier paper (Hunter, 2002) it was shown that mean first passage times play an important role in determining bounds on the relative and absolute differences between the stationary probabilities in perturbed finite irreducible discrete time Markov chains. Further when two perturbations of the transition probabilities in a single row are carried out the differences between the stationary probabilities in the unperturbed and perturbed situations are easily expressed in terms of a reduced number of mean first passage times. Using this procedure we provide an updating procedure for mean first passage times to determine changes in the stationary distributions under successive perturbations. Simple procedures for determining both stationary distributions and mean first passage times in a finite irreducible Markov chain are also given. The techniques used in the paper are based upon the application of generalized matrix inverses.
