Some asymptotic results for transient random walks

We consider a real-valued random walk S which drifts to -infinity and is such that E(exp theta S-1)<infinity for some theta>0, but for which Cramer'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.