Random Walks Crossing High Level Curved Boundaries

Let {S_n} be a random walk, generated by i.i.d. increments X_i, which drifts weakly to ∞ in the sense that S_n →^P ∞ 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 → an^K before it crosses the curve n → - an^K tends to 1 as a → ∞. This intuitively plausible result is not true for K = 1, however, and for 1/2 < K < 1, the converse results are not true in general, either. More general boundaries g(n) than g(n) = n^K are also considered, and we also prove similar results for first passages out of regions like {(n, y): n ≥ 1, \y\ ≤ (a + n)^K} as a → ∞.