PRECISE ASYMPTOTICS IN COMPLETE MOMENT CONVERGENCE FOR DEPENDENT RANDOM VARIABLE

• Journal title : Honam Mathematical Journal
• Volume 31, Issue 3,  2009, pp.369-380
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2009.31.3.369
Title & Authors
PRECISE ASYMPTOTICS IN COMPLETE MOMENT CONVERGENCE FOR DEPENDENT RANDOM VARIABLE
Han, Kwang-Hee;

Abstract
Let $\small{X,X_1,X_2,\;{\cdots}}$ be identically distributed and negatively associated random variables with mean zeros and positive, finite variances. We prove that, if $\small{E{\mid}X_1{\mid}^r}$ < $\small{{\infty}}$, for 1 < p < 2 and r > $\small{1+{\frac{p}{2}}}$, and $\small{lim_{n{\rightarrow}{\infty}}n^{-1}ES^2_n={\sigma}^2}$ < $\small{{\infty}}$, then $\small{lim_{{\epsilon}{\downarrow}0}{\epsilon}^{{2(r-p}/(2-p)-1}{\sum}^{\infty}_{n=1}n^{{\frac{r}{p}}-2-{\frac{1}{p}}}E\{{{\mid}S_n{\mid}}-{\epsilon}n^{\frac{1}{p}}\}+={\frac{p(2-p)}{(r-p)(2r-p-2)}}E{\mid}Z{\mid}^{\frac{2(r-p)}{2-p}}}$, where $\small{S_n\;=\;X_1\;+\;X_2\;+\;{\cdots}\;+\;X_n}$ and Z has a normal distribution with mean 0 and variance $\small{{\sigma}^2}$.
Keywords
Precise asymptotics;Complete moment convergence;Negatively associated;Berry-Esseen inequality;
Language
English
Cited by
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