A probabilistic proof of Cooper and Frieze's "First Visit Time Lemma"

TL;DR

This paper offers a new probabilistic proof of the First Visit Time Lemma for Markov chains, simplifying previous complex analysis methods and providing quantitative bounds under standard assumptions.

Contribution

It introduces a probabilistic approach to prove the FVTL, avoiding complex analysis and extending the understanding of hitting times in Markov chains.

Findings

Probabilistic proof of FVTL using quasi-stationary distributions

Quantitative bounds on Doob's transform of the chain

Exponential decay of hitting time distribution under assumptions

Abstract

In this short note we present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in [21], and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on $n$ states. We assume that the chain is rapidly mixing and with a stationary measure having no entry which is too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state $x$ -- for the chain started at stationarity -- up to a small multiplicative correction. While the proof of the FVTL presented by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob's transform of the chain on the complement of the state $x$.