Applications of Paz's inequality to perturbation bounds for Markov chains (Q1375093)

Let \(T\) and \(T+E\) be two \(n\times n\) nonnegative stochastic irreducible transition matrices for Markov processes whose stationary distribution vectors are \(\pi\) and \(\tilde \pi\), resp. The total error \(|\pi - \tilde \pi|_\infty\) and the relative errors \(|(\pi_j - \tilde \pi_j)/\pi_j|\) of the components can be estimated in terms of the disturbance \(E\) with coefficients which represent a sort of 'condition number' of \(A\). Using an inequality of \textit{A. Paz} [Introduction to probabilistic automata (1971; Zbl 0234.94055), Chapter IIa] the authors improve known estimates of this kind. The improvement is roughly by a factor of 2.