ABSTRACT: Uniform Stability Of Markov Chains

$Carl Meyer$

Uniform Stability Of Markov Chains

ABSTRACT: By deriving a new set of tight perturbation bounds, it is shown that all stationary probabilities of a finite irreducible Markov chain react essentially in the same way to perturbations in the transition probabilities. In particular, if at least one stationary probability is insensitive in a relative sense, then all stationary probabilities must be insensitive in an absolute sense. New measures of sensitivity are related to more traditional ones, and it is shown that all relevant condition numbers for the Markov chain problem are small multiples of each other. Finally, the implications of these findings to the computation of stationary probabilities by direct methods are discussed, and the results are applied to stability issues in nearly transient chains. Prof. Ipsen's work was supported in part by National Science Foundation grant CCR-9102853, and Prof. Meyer's work was supported in part by National Science Foundation grants DMS-9020915 and DDM-8906248.

JOURNAL: SIAM J. Matrix Anal. Appl.; Vol. 15, No. 4, October, 1994, pp. 1061-1074.

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