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

The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk