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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 39, Fasc. 2,
pages 19 - 38
DOI: 10.19195/0208-4147.39.1.2
 

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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