On the asymptotic behaviour of first passage times for transient random walk (Q1102039)

Let \(\tau _ x\) denote the time at which a random walk with finite positive mean first passes into (x,\(\infty)\), where \(x\geq 0\). This paper establishes the asymptotic behaviour of \(\Pr \{\tau _ x>n\}\) as \(n\to \infty\) for fixed x 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 \(\lim _{n\to \infty}\Pr \{\tau _ n>n\}/\Pr \{\tau _ 0>n\}\) coincides with the limit of the same quantity for recurrent random walks satisfying Spitzer's condition, but in the second case the limit is more complicated.