Volume 14, Number 1, Spring 2006
A REMARK ON THE PRECISE ASYMPTOTICS
IN THE BAUMKATZ LAWS OF
LARGE NUMBERS
FENGYANG CHENG AND YUEBAO WANG
Abstract. Let {X, X_{i} : i ≥ 1} be a sequence of i.i.d.
random variables with common distribution function F(x) and
let S_{n} = X_{i}, 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 (S_{n} — a_{n} )/ b_{n} )
G, as n → ∞ for suitable a_{n} and b_{n}. 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 n_{0} and nonnegative function h(x),
where Z is a random variable having the distribution G.
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