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.