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

  1. Shuhua Chang
  2. Deli Li
  3. Andrew Rosalsky

Abstract

 

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.

 

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Probability and Mathematical Statistics

39, z. 1, 2019

Pages from 19 to 38

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