Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + … + Xn, n ≥ 1. Motivated by a theorem of Mikosch (1984), this note is devoted to establishing a strong law of large numbers for the sequence {max1 ≤ k ≤ n |Sk|; n ≥ 1}. More specifically, necessary and sufficient conditions are given for [wzór] a.s., where log x = loge max{e, x}, x ≥ 0.