The upper limit of a normalized random walk

We consider the upper limit of S-n/n(1/p) where S-n are partial sums of iid random variables. Under the assumption of E(X+)(P) < infinity, we provide an integral test which determines the upper limit up to certain universal constant factors depending on p only. The problem is closely related to moment properties of ladder variables. We prove our theorem by considering the lower limit of T-k/k(p) where Tk is the k-th epoch of the random walk.