Abstract.
We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup n →∞(S n /nκ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.
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This work is partially supported by ARC Grant DP0210572.
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Doney, R., Maller, R. Passage times of random walks and Lévy processes across power law boundaries. Probab. Theory Relat. Fields 133, 57–70 (2005). https://doi.org/10.1007/s00440-004-0414-3
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DOI: https://doi.org/10.1007/s00440-004-0414-3