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Improved bounds for a condition number for Markov chains. (English) Zbl 1052.65004
Let $P$ and $\widetilde P=P+E$ be transition matrices of some finite ($n$ states) homogeneous ergodic Markov chains, $\pi$ and $\widetilde\pi$ be their stationary distributions. The article is devoted to the condition number $\kappa_4$, i.e. the constant for which $$|\pi-\widetilde\pi|_\infty\le\kappa_4|E|_\infty.$$ New upper bounds for $\kappa_4$ are obtained in terms of $Q=(q_{ij})_{i,j=1}^n=I-P$ and its group inverse $Q^{\#}$. E.g. $$ \kappa_4\le { \delta_2+\sigma_2\delta_3+\dots+\sigma_{n-2}\delta_{n-1}+\sigma_{n-1} \over \chi }, $$ where $$ \sigma_k=\max_{j_1,j_2}(q_{j_2,j_1}+q_{j_2,j_2})\cdots \max_{j_1\dots j_k}(q_{j_k,j_1}+\dots+q_{j_k,j_k}), $$ $$ \delta_k=\max_{j_1\dots j_k} {\prod_{i=1}^n q_{ii}\over q_{j_1j_1}\dots q_{j_kj_k}}, \chi=\prod_{\lambda_j\not=1}(1-\lambda_j), $$ $\lambda_j$ are eigenvalues of $P$. Numerical results show that these bounds are approximately two times better than the estimate of {\it C. D. Meyer} for $n=10$ [SIAM J. Matrix. Anal. Appl. 15, No. 3, 715--728 (1994; Zbl 0809.65143)].

MSC:
65C40Computational Markov chains (numerical analysis)
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
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