On optimal condition numbers for Markov chains

Let T and T̃ = T −E be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)–norm pair if ‖π − π̃‖α ≤ κ(T )‖E‖β . Here π and π̃ are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect to the (1,∞) and (∞,∞)–norm pairs have been suggested in the literature. They were ranked according to their size by Cho and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient τp(A #) = supyte=0 ‖ytA#‖p ‖y‖1 , where e is the n–vector of all 1’s, is the smallest of the condition number of Markov chains with respect to the (p,∞)–norm pair. We use this result to identify the smallest of the condition numbers of Markov chains among the (∞,∞) and (1,∞)–norm pairs. These are, respectively, κ3 and κ6 in the Cho–Meyer list of 8 condition numbers. Kirkland has studied κ3(T ). He has shown that κ3(T ) ≥ n−1 2n and he has characterized transition matrices for which equality holds. We prove here again that 2κ3(T ) ≤ κ(6) which appears in the Cho–Meyer paper and we characterize the transition matrices T for which κ6(T ) = n−1 n . There is actually only one such matrix: T = (Jn − I)/(n − 1), where Jn is the n× n matrix of all 1’s.