Uniform Stability Of Markov Chains
- 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.
- SIAM J. Matrix Anal. Appl.
- Vol. 15, No. 4, October, 1994, pp. 1061-1074.
- Ilse C. F. Ipsen
- Carl D. Meyer
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