On the asymptotic behaviour of first passage times for transient random walk

SummaryLet τx denote the time at which a random walk with finite positive mean first passes into (x, ∞), wherex≧0. This paper establishes the asymptotic behaviour of Pr {τx >n} asn→∞ for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case $$\mathop {\lim }\limits_{n \to \infty } $$ Pr {τx >n}/Pr{τ0 >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.