On the Kemeny constant and stationary distribution vector for a Markov chain

Abstract

Suppose that A is an irreducible stochastic matrix of order n, and denote its eigenvalues by 1, λ2, . . . , λn. The Kemeny constant, K(A) for the Markov chain associated with A is defined as K(A) = ∑n j=2 1 1−λj , and can be interpreted as the mean first passage from an unknown initial state to an unknown destination state in the Markov chain. Let w denote the stationary distribution vector for A, and suppose that w1 ≤ w2 ≤ · · · ≤ wn. In this paper, we show that K(A) ≥ ∑n j=1(j − 1)wj , and we characterise the matrices yielding equality in that bound. The results are established using techniques from matrix theory and the theory of directed graphs.