Regular Markov chains for which the transition matrix has large exponent

Abstract

In this paper we analyze properties of transition matrices T∈ R n,n of regular Markov chains whose exponent (index of primitivity) is bounded below by ⌊((n−1) 2+1)/2⌋+2 . Our investigation leads to a proof that for such a T, the corresponding condition number K(T):= max 1⩽i,j⩽n|Q # i,j|<3/2 , where Q=I−T. We go on to show that if E∈ R n,n is a perturbation of T such that T+E is also a transition matrix for a regular Markov chain and π and π ̃ are the stationary distribution vectors of T and T+E, respectively, then ∥π− π ̃ ∥ ∞<(5/4)∥E∥ ∞ and ∥π− π ̃ ∥ 1<((3n+3)/4)∥E∥ 1 . Many of the techniques used to establish these results are combinatorial in nature.