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 < ∞, 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 T_k is the k-th epoch of the random walk.