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Improved bounds for a condition number for Markov chains. (English) Zbl 1052.65004

Let $P$ and $\stackrel{˜}{P}=P+E$ be transition matrices of some finite ($n$ states) homogeneous ergodic Markov chains, $\pi$ and $\stackrel{˜}{\pi }$ be their stationary distributions. The article is devoted to the condition number ${\kappa }_{4}$, i.e. the constant for which

$|\pi -\stackrel{˜}{\pi }{|}_{\infty }\le {\kappa }_{4}{|E|}_{\infty }·$

New upper bounds for ${\kappa }_{4}$ are obtained in terms of $Q={\left({q}_{ij}\right)}_{i,j=1}^{n}=I-P$ and its group inverse ${Q}^{#}$. E.g.

${\kappa }_{4}\le \frac{{\delta }_{2}+{\sigma }_{2}{\delta }_{3}+\cdots +{\sigma }_{n-2}{\delta }_{n-1}+{\sigma }_{n-1}}{\chi },$

where

${\sigma }_{k}=\underset{{j}_{1},{j}_{2}}{max}\left({q}_{{j}_{2},{j}_{1}}+{q}_{{j}_{2},{j}_{2}}\right)\cdots \underset{{j}_{1}\cdots {j}_{k}}{max}\left({q}_{{j}_{k},{j}_{1}}+\cdots +{q}_{{j}_{k},{j}_{k}}\right),$
${\delta }_{k}=\underset{{j}_{1}\cdots {j}_{k}}{max}\frac{{\prod }_{i=1}^{n}{q}_{ii}}{{q}_{{j}_{1}{j}_{1}}\cdots {q}_{{j}_{k}{j}_{k}}},\chi =\prod _{{\lambda }_{j}\ne 1}\left(1-{\lambda }_{j}\right),$

${\lambda }_{j}$ are eigenvalues of $P$.

Numerical results show that these bounds are approximately two times better than the estimate of C. D. Meyer for $n=10$ [SIAM J. Matrix. Anal. Appl. 15, No. 3, 715–728 (1994; Zbl 0809.65143)].

##### MSC:
 65C40 Computational Markov chains (numerical analysis) 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 65F35 Matrix norms, conditioning, scaling (numerical linear algebra)