A note on asymptotics of linear combinations of iid random variables |
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Authors: | Péter Kevei |
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Affiliation: | 1.Analysis and Stochastics Research Group of the Hungarian Academy of Sciences Bolyai Institute,University of Szeged,Szeged,Hungary;2.Centro de Investigación en Matemáticas,Guanajuato,Mexico |
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Abstract: | Let X1,X2, ... be iid random variables, and let a
n
= (a
1,n, ..., a
n,n
) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination $
S_{a_n }
$
S_{a_n }
= a
1,n
X
1 + ... + a
n,n
X
n
under the natural negligibility condition lim
n→∞
max{|a
k,n
|: k = 1, ..., n} = 0. We prove that if $
S_{a_n }
$
S_{a_n }
is asymptotically normal for a weight sequence a
n
, in which the components are of the same magnitude, then the common distribution belongs to $
\mathbb{D}
$
\mathbb{D}
(2). |
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Keywords: | |
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