Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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ON PRECISE ASYMPTOTICS IN THE LAW OF LARGE NUMBERS OF ASSOCIATED RANDOM VARIABLES

  • Baek, Jong-Il (School of Mathematics and Informational Statistics, Wonkwang University) ;
  • Seo, Hye-Young (School of Mathematics and Informational Statistics, Wonkwang University) ;
  • Lee, Gil-Hwan (School of Mathematics and Informational Statistics, Wonkwang University)
  • Received : 2007.08.27
  • Accepted : 2008.02.13
  • Published : 2008.03.25
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Abstract

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

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References

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