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In this short note we present an alternative proof of the so-called FirstVisit Time Lemma (FVTL), originally presented by Cooper and Frieze in its firstformulation 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 growingsequence of irreducible Markov chains on $n$ states. We assume that the chainis rapidly mixing and with a stationary measure having no entry which is toosmall nor too large. Under these assumptions, the FVTL shows the exponentialdecay of the distribution of the hitting time of a given state $x$ -- for thechain started at stationarity -- up to a small multiplicative correction. Whilethe proof of the FVTL presented by Cooper and Frieze is based on tools fromcomplex analysis, and it requires an additional assumption on a generatingfunction, we present a completely probabilistic proof, relying on the theory ofquasi-stationary distributions and on strong-stationary times arguments. Inaddition, under the same set of assumptions, we provide some quantitativecontrol on the Doob's transform of the chain on the complement of the state$x$.

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