Tight inequalities among set hitting times in Markov chains

Abstract:

Given an irreducible discrete time Markov chain on a finite state space, we consider the largest expected hitting time $T(\alpha )$ of a set of stationary measure at least $\alpha$ for $\alpha \in (0,1)$. We obtain tight inequalities among the values of $T(\alpha )$ for different choices of $\alpha$. One consequence is that $T(\alpha ) \le T(1/2)/\alpha$ for all $\alpha < 1/2$. As a corollary we have that if the chain is lazy in a certain sense as well as reversible, then $T(1/2)$ is equivalent to the chain’s mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions $T(\alpha )$ over the domain $\alpha \in (0,1/2]$.