Random walks crossing power law boundaries

We collect together some known results, and prove some new results, giving criteria for lim sup/n→∞ |S_n|/n^κ = ∞ a.s. or lim sup/n→∞ S_n/n^κ = ∞ a.s., where S_n is a random walk and K ≧ 0. Conditions which are necessary and sufficient are given for all cases, and the conditions are quite explicit in all but one case (the case 1/2 < K < 1, E|X| < ∞, EX = 0 for lim sup/n→∞ S_n/n^k). The results are related to the finiteness of the first passage times of the random walk out of the regions {(n, y) : n ≧ 1, |y| ≦ an^κ} and {(n, y) : n ≧1, y ≦ an^κ}, where K > 0, a > 0.