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Contents of PMS, Vol. 39, Fasc. 1,
pages 19 - 38
 

STRONG LAWS OF LARGE NUMBERS FOR THE SEQUENCE OF THE MAXIMUM OF PARTIAL SUMS OF I.I.D. RANDOM VARIABLES

Shuhua Chang
Deli Li
Andrew Rosalsky

Abstract: Let 0 < p ≤ 2 , let (X  ; n ≥ 1)
   n be a sequence of independent copies of a real-valued random variable X , and set S = X  + ...+ X
 n    1        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      |S |; n ≥ 1)
    1≤k≤n  k . More specifically, necessary and sufficient conditions are given for

lim  ( max |S |)(logn)-1 = e1∕p a.s.,
n→∞ 1≤k≤n  k

where logx = logemax (e,x) , x ≥ 0 .

2000 AMS Mathematics Subject Classification: Primary: 60F15; Secondary: 60G50, 60G70.

Keywords and phrases: Theorem of Mikosch, i.i.d. real-valued random variables, maximum of partial sums, strong law of large numbers.

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