Abstract: | Let {S
n} be a random walk, generated by i.i.d. increments X
i which drifts weakly to in the sense that
as n . Suppose k 0, k 1, and E|X
1|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 . |