Random walks crossing power law boundaries

We collect together some known results, and prove some new results, giving criteria for lim(n-->infinity) sup \S-n\/n(kappa) = infinity a.s. or lim(n- ->infinity) sup S-n/(kappa)(n) = infinity a.s., where S-n is a random walk and kappa greater than or equal to 0. Conditions which are necessary and su fficient are given for all cases, and the conditions are quite explicit in all but one case (the case 1/2 < kappa < 1, E\X\ < infinity, EX = 0 for lim (n-->infinity) sup S-n/(kappa)(n)). The results are related to the finitene ss of the first passage times of the random walk out of the regions {(n, y) : n greater than or equal to 1, \y\ less than or equal to an(kappa)} and { (n, y) : n greater than or equal to 1, y less than or equal to an(kappa)}, where kappa > 0, a > 0.