Sensitivity of finite Markov chains under perturbation (Q1802441)

Let \(P\) and \(\overline{P}\) be finite stochastic matrices containing only one irreducible set of states, and \(\pi^ T\), \(\overline {\pi}^ T\) their corresponding unique stationary distribution vectors. Put \(E=\overline {P}-P\) and denote by \(A^ \#\) the group generalized inverse of \(A=I-P\). In a previous paper, the author proved that \(\|\pi^ T- \overline{\pi}^ T\|_ 1\leq \| E\|_ 1\tau_ 1(A^ \#)\), where \(\tau_ 1(B)={1\over 2}\max_{i,j} \sum^ n_{s=1} | B_{is}-B_{js}|\) is the ergodicity coefficient of the (stochastic) matrix \(B\). As a consequence, \(\tau_ 1(A^ \#)\) can be viewed as a relative sensitivity factor under perturbation. The aim of this paper is to find bounds for \(\tau_ 1(A^ \#)\) in terms of the non-unit eigenvalues of \(P\).