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

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 [21], then used refined a list papers Cooper, coauthors. We work original setting, considering growing sequence irreducible Markov chains on $n$ states. assume that chain is rapidly mixing with stationary measure having no entry which too small nor large. Under these assumptions, FVTL shows exponential decay distribution hitting time given state $x$ -- for started at stationarity up to multiplicative correction. While based tools from complex analysis, it requires additional assumption generating function, completely probabilistic proof, relying theory quasi-stationary distributions strong-stationary times arguments. addition, under same set provide some quantitative control Doob's transform complement $x$.