Periodica Mathematica Hungarica
Volume/Issue:
Volume 60: Issue 1
A note on asymptotics of linear combinations of iid random variables
Author: Péter Kevei
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  • 1 University of Szeged Analysis and Stochastics Research Group of the Hungarian Academy of Sciences Bolyai Institute Aradi vértanúk tere 1 H-6720 Szeged Hungary
  • 2 Centro de Investigación en Matemáticas Callejón Jalisco S/N, Mineral de Valenciana Guanajuato 36240 Mexico
DOI:
https://doi.org/10.1007/s10998-010-1025-7
Page Count:
25–36
Publication Date:
01 Mar 2010
Online Publication Date:
17 Mar 2010
Article Category:
Research Article
Restricted access

Abstract  

Let X1,X2, ... be iid random variables, and let an = (a1,n, ..., an,n) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_{a_n }$$ \end{document}
= a1,nX1 + ... + an,nXn under the natural negligibility condition limn→∞ max{|ak,n|: k = 1, ..., n} = 0. We prove that if
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_{a_n }$$ \end{document}
is asymptotically normal for a weight sequence an, in which the components are of the same magnitude, then the common distribution belongs to
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{D}$$ \end{document}
(2).

Cover Periodica Mathematica Hungarica
Periodica Mathematica Hungarica
Print ISSN:
0031-5303
Online ISSN:
1588-2829
Keywords:
60F05; linear combinations; asymptotic normality; domain of attraction

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