Some asymptotic results for transient random walks

The authors consider a real-valued random walk S which drifts to ¸-• and is such that E(expθS1)<• for some θ>0, but for which Cramér’s condition fails. They 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, the authors obtain new limit theorems for certain conditional laws.