Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables
DOI:
https://doi.org/10.19195/0208-4147.39.1.2Słowa kluczowe:
Theorem of Mikosch, i.i.d. real-valued random variables, maximum of partial sums, strong law of large numbersAbstrakt
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
limn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,
where log x = loge max{e, x}, x ≥ 0.