Some asymptotic results for transient random walks

Abstract

We consider a real-valued random walk S which drifts to –∞ and is such that E(exp θS 1) < ∞ for some θ > 0, but for which Cramér's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws.