Convergence theory of some classes of iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices

Abstract

This contribution is a natural follow-up of the paper of the same authors entitled Convergence theory of an aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices published in [Numer. Linear Algebra Appl. 5 (1998) 253]. In contrast to that paper in which the algorithm studied was based on the splitting whose iteration matrix was identical with the matrix whose stationary probability vectors are computed, the present paper presents a convergence analysis of algorithms based on fully general splittings of nonnegative type. Together with this generalization another feature of the older paper and namely the independence of the convergence results on the size of the elements of the examined stochastic matrix is shown to remain valid for the new algorithms as well. This concerns in particular the possibility of computing stationary probability vectors of Markov chains containing rare events, i.e. events whose stationary probabilities are substantially smaller than some of elements of the transition matrix of the chain.