ON PRECISE ASYMPTOTICS IN THE LAW OF LARGE NUMBERS OF ASSOCIATED RANDOM VARIABLES

• Journal title : Honam Mathematical Journal
• Volume 30, Issue 1,  2008, pp.9-20
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2008.30.1.009
Title & Authors
ON PRECISE ASYMPTOTICS IN THE LAW OF LARGE NUMBERS OF ASSOCIATED RANDOM VARIABLES
Baek, Jong-Il; Seo, Hye-Young; Lee, Gil-Hwan;

Abstract
Let $\small{{X_i{\mid}i{\geq}1}}$ be a strictly stationary sequence of associated random variables with mean zero and let $\small{{\sigma}^2=EX_1^2+2\sum\limits_{j=2}^\infty{EX_1}{X_j}}$ with 0 < $\small{{\sigma}^2}$ < $\small{{\infty}}$. Set $\small{S_n={\sum\limits^n_{i=1}^\{X_i}}$, the precise asymptotics for $\small{{\varepsilon}^{{\frac{2(r-p)}{2-p}}-1}\sum\limits_{n{\geq}1}n^{{\frac{r}{p}}-{\frac{1}{p}}+{\frac{1}{2}}}P({\mid}S_n{\mid}{\geq}{\varepsilon}n^{{\frac{1}{p}}})}$,$\small{{\varepsilon}^2\sum\limits_{n{\geq}3}{\frac{1}{nlogn}}p({\mid}Sn{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})}$ and $\small{{\varepsilon}^{2{\delta}+2}\sum\limits_{n{\geq}1}{\frac{(loglogn)^{\delta}}{nlogn}}p({\mid}S_n{\mid}{\geq}{\varepsilon\sqrt{nloglogn}})}$ as $\small{{\varepsilon}{\searrow}0}$ are established under the suitable conditions.
Keywords
Associated random variables;Law of large numbers;Precise asymptotics;
Language
English
Cited by
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