Abstract
An algorithmic procedure for the determination of the stationary distribution of a finite, m‐state, irreducible Markov chain, that does not require the use of methods for solving systems of linear equations, is presented. The technique is based upon a succession of m, rank one, perturbations of the trivial doubly stochastic matrix whose known steady state vector is updated at each stage to yield the required stationary probability vector.
Highlights
Widespread attention has been given to the computation of stationary distributions of Markov chains
Suppose that following the th perturbation, the stationary probability vector r has been found for the Markov chain with transition matrix Pi, as given by (1.1), by using the procedure u described by Theorem 2.1 (b) for suitable choices of and i"
A consequence of this is that the effect of changing selected transition probabilities upon the stationary distribution can be determined. (See Hunter (1986))
Summary
Widespread attention has been given to the computation of stationary distributions of Markov chains. Paige, Styan and Wachter (1975) presented a comprehensive survey of eight different algorithms involving a variety of procedures including the use of generalized inverses, rank reduction, least squares and power methods Their recommendation was a direct method that involved transforming the singular set of stationary equations into a non-singular system using a rank one modification followed by Gaussian elimination with row pivoting. In Hunter (1986), techniques for updating the stationary distribution of a finite irreducible Markov chain, following a rank one perturbation of its transition matrix, were presented. In this current paper, these techniques are utilized, to construct a general procedure for determining the stationary distribution of any finite irreducible Markov chain.
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