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

摘要

Let 0 p ? 2, let {Xn; n - 1} be a sequence of independentcopies of a real-valued random variable X, and set Sn = X1 +. . . + Xn, n - 1. Motivated by a theorem of Mikosch (1984), this noteis devoted to establishing a strong law of large numbers for the sequence{max1?k?n |Sk| ; n - 1}. More specifically, necessary and sufficient conditionsare given forlimn!1( max1?k?n|Sk|)(log n)?1= e1/p a.s.,where log x = loge max{e, x}, x - 0.