The Tail of the Length of an Excursion in a Trap of Random Size

Abstract

AbstractConsider a random walk with a drift to the right on$$\{0,\ldots ,k\}$${0,…,k}wherekis random and geometrically distributed. We show that the tail$${\mathbb {P}}[T>t]$$P[T>t]of the lengthTof an excursion from 0 decreases up to constants like$$t^{-\varrho }$$t-ϱfor some$$\varrho >0$$ϱ>0but is not regularly varying. We compute the oscillations of$$t^\varrho \,{\mathbb {P}}[T>t]$$tϱP[T>t]as$$t\rightarrow \infty $$t→∞explicitly.