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Volume 14,     Number 1,     Spring 2006

 

A REMARK ON THE PRECISE ASYMPTOTICS
IN THE BAUM-KATZ LAWS OF
LARGE NUMBERS
FENG-YANG CHENG AND YUE-BAO WANG

Abstract. Let {X, Xi : i ≥ 1} be a sequence of i.i.d. random variables with common distribution function F(x) and let Sn = Xi, n ≥ 1. Suppose that F belongs to the domain of attraction of a nondegenerate stable distribution G with characteristic exponent α (0 < α ≤ 2), i.e., that (Snan )/ bn ) G, as n → ∞ for suitable an and bn. Let g(x) (defined on [0, + ∞)) be an increasing functions such that g-1(x) is a regular varying function with index ρ(0 ≤ ρ < α ). We prove that
                      
for some positive integer n0 and nonnegative function h(x), where Z is a random variable having the distribution G.

 

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© 2006, Canadian Applied Mathematics Quarterly (CAMQ)