Dual bounds on the equilibrium distribution of a finite Markov chain (Q1092519)

Consider a stationary finite Markov chain with states numbered 1,2,...,N and transition probability matrix \(P=[P_{ij}]\), \(1\leq i\), \(j\leq N\), \(P_{ij}\geq 0\), \(\sum^{N}_{j=1}P_{ij}=1\). The author assumes that P is both unichained and aperiodic and gives sharp upper and lower bounds for the equilibrium state probabilities \(\pi_ k\) (1\(\leq k\leq N)\) of P, along with an iterative scheme which moves the bounds monotonically and geometrically inward to \(\pi_ k.\) The present bounds generalize a result of \textit{J. G. Kemeny} and \textit{J. L. Snell} [Finite Markov Chains (p.173) (1960; Zbl 0089.137)]. It may be noted that they have the advantage of being tight for small values of n, thereby reducing computational effort, but they have the disadvantage of working for only one choice of k at a time, so that bounding the entire equilibrium distribution requires N-fold effort.