  We study the asymptotic probability that a random walk with heavy-tailed
increments crosses a high boundary on a random time interval. We use new
techniques to extend results of Asmussen [Ann. Appl. Probab. 8 (1998) 354-374]
to completely general stopping times, uniformity of convergence over all
stopping times and a wide class of nonlinear boundaries. We also give some
examples and counterexamples.
