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

Chang Shuhua, Li Deli, Rosalsky Andrew

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 1

19--38

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.