Sensitivity Of The Stationary Distribution Of A Markov Chain
- It is well known that if the transition matrix of an
irreducible Markov chain of moderate size has a subdominant
eigenvalue which is close to 1, then the chain is
ill conditioned in the sense that there are stationary probabilities
which are sensitive to perturbations in the transition
probabilities. However, the converse of this statement has heretofore
been unresolved. The purpose of this article is to address this issue
by establishing upper and lower bounds on the condition number of
the chain such that the bounding terms are functions of the
eigenvalues of the transition matrix. Furthermore, it is demonstrated
how to obtain estimates for the condition number of an irreducible
chain with little or no extra computational effort over that required
to compute the stationary probabilities by means of an LU or QR
factorization. This work was supported in part by National
Science Foundation grants DMS-9020915 and DDM-8906248.
- SIAM J. Matrix Anal. Appl.
- Vol. 15, No. 3, July, 1994, pp. 715-728
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