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Digraph-based conditioning for Markov chains. (English) Zbl 1056.15022
For an irreducible stochastic matrix $T$, the author considers a certain condition number $c(T)$, which measures the stability of the corresponding stationary distribution when $T$ is perturbed, and then characterizes the strongly connected directed graphs $D$ such that $c(T)$ is bounded as $T$ ranges over $\Cal S_D$, the set of stochastic matrices whose directed graph is contained in $D$. For those digraphs $D$ for which $c(T)$ is bounded, the maximum value of $\{c(T)\mid T$ ranges over $\Cal S_D\}$ is determined.

15B51Stochastic matrices
15A18Eigenvalues, singular values, and eigenvectors
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
05C20Directed graphs (digraphs), tournaments
15A12Conditioning of matrices
Full Text: DOI
[1] Brualdi, R.; Ryser, H.: Combinatorial matrix theory. (1991) · Zbl 0746.05002
[2] Campbell, S.; Meyer, C.: Generalized inverses of linear transformations. (1991) · Zbl 0732.15003
[3] Cho, G.; Meyer, C.: Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear algebra appl. 335, 137-150 (2001) · Zbl 0983.60062
[4] Haviv, M.; Van Der Heyden, L.: Perturbation bounds for the stationary probabilities of a finite Markov chain. Adv. in appl. Prob. 16, 804-818 (1984) · Zbl 0559.60055
[5] Kirkland, S.: The group inverse associated with an irreducible periodic nonnegative matrix. SIAM J. Matrix anal. Appl. 16, 1127-1134 (1995) · Zbl 0838.15003
[6] Kirkland, S.: A note on the eigenvalues of a primitive matrix with large exponent. Linear algebra appl. 253, 103-112 (1997) · Zbl 0878.15015
[7] Kirkland, S.: On a question concerning condition numbers for Markov chains. SIAM J. Matrix anal. Appl. 23, 1109-1119 (2002) · Zbl 1013.15005
[8] Kirkland, S.: Conditioning properties of the stationary distribution for a Markov chain. Electron. J. Linear algebra 10, 1-15 (2003) · Zbl 1022.15027
[9] Kirkland, S.; Neumann, M.: Regular Markov chains for which the transition matrix has large exponent. Linear algebra appl. 316, 45-65 (2000) · Zbl 0965.15026
[10] Kirkland, S.; Neumann, M.; Shader, B.: Applications of paz’s inequality to perturbation bounds for Markov chains. Linear algebra appl. 268, 183-196 (1998) · Zbl 0891.65147
[11] Meyer, C.: The condition of a finite Markov chain and perturbation bounds for the limiting probabilities. SIAM J. Discrete and algebraic methods 1, 273-283 (1980) · Zbl 0498.60071
[12] Seneta, E.: Non-negative matrices and Markov chains. (1981) · Zbl 0471.60001