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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 lim n→∞ ( max 1¬k¬n |Sk|) −1 = e 1/p a.s., where log x = loge max{e, x}, x ­ 0. 2010 AMS Mathematics Subject Classification: Primary: 60F15; Secondary: 60G50, 60G70.

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