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

A telegraph process with an elastic barrier at the origin was studied in [5]; 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 [17]. In this paper we study large and moderate deviations for a generalized model where M is a light-tailed distributed random variable.