Asymptotic results for the absorption time of telegraph processes with a non-standard barrier at the origin

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A telegraph process with an elastic barrier at the origin was studied in Di Crescenzo et al. (2018); in particular the number of visits of the origin before the absorption is a geometric distributed random variable M. Some asymptotic results (large and moderate deviations) for that model were obtained in Macci et al. (2021). In this paper we study large and moderate deviations for a generalized model where M is a light-tailed distributed random variable.