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

Abstrakt

Let 0 < p ≤ 2, let {X_n; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set S_n = X₁ + . . . + X_n, n ≥ ­ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max_1≤k≤n |S_k| ; n ≥ ­ 1}. More specifically, necessary and sufficient conditions are given for

lim_n→∞ max_1≤k≤n |S_k|^{log n−1} = e^1/p a.s.,

where log x = log_e max{e, x}, x ≥­ 0.