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 ( α ) T(\alpha ) of a set of stationary measure at least α \alpha for α ∈ ( 0 , 1 ) \alpha \in (0,1) . We obtain tight inequalities among the values of T ( α ) T(\alpha ) for different choices of α \alpha . One consequence is that T ( α ) ≤ T ( 1 / 2 ) / α T(\alpha ) \le T(1/2)/\alpha for all α > 1 / 2 \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 ) 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 ( α ) T(\alpha ) over the domain α ∈ ( 0 , 1 / 2 ] \alpha \in (0,1/2] .