Random Walks Crossing High Level Curved Boundaries

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Let {Sn} be a random walk, generated by i.i.d. increments Xi which drifts weakly to ∞ in the sense that Sn r → ∞ as n→ ∞. Suppose k≥0, k≠1, and E|X1|1\k = ∞ if k>1. Then we show that the probability that S. crosses the curve n↦anK before it crosses the curve n ↦ −ank tends to 1 as a → ∞. This intuitively plausible result is not true for k = 1, however, and for 1/2

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