Tight inequalities among set hitting times in Markov chains

TL;DR

This paper establishes tight inequalities among expected hitting times for Markov chains, revealing relationships with mixing times and providing a limiting characterization of hitting time functions.

Contribution

It derives optimal inequalities among hitting times for different stationary measure thresholds, linking them to mixing times in reversible chains and characterizing their limiting behavior.

Findings

T(eta)  T(eta)/eta for eta<1/2

T(1/2) is equivalent to the chain's mixing time under certain conditions

Provides a pointwise limiting characterization of T() functions

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 $\alpha\in(0,1/2]$.