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

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

|π-π ˜| κ 4 |E| ·

New upper bounds for κ 4 are obtained in terms of Q=(q ij ) i,j=1 n =I-P and its group inverse Q # . E.g.

κ 4 δ 2 +σ 2 δ 3 ++σ n-2 δ n-1 +σ n-1 χ,


σ k =max j 1 ,j 2 (q j 2 ,j 1 +q j 2 ,j 2 )max j 1 j k (q j k ,j 1 ++q j k ,j k ),
δ k =max j 1 j k i=1 n q ii q j 1 j 1 q j k j k ,χ= λ j 1 (1-λ j ),

λ 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)].

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