Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A SUPPLEMENT TO PRECISE ASYMPTOTICS IN THE LAW OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED SUMS

  • Hwang, Kyo-Shin (THE RESEARCH INSTITUTE OF NATURAL SCIENCE GYEONGSANG NATIONAL UNIVERSITY)
  • Published : 2008.11.01
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Abstract

Let X, $X_1$, $X_2$, ... be i.i.d. random variables with zero means, variance one, and set $S_n\;=\;{\sum}^n_{i=1}\;X_i$, $n\;{\geq}\;1$. Gut and $Sp{\check{a}}taru$ [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P($|S_n|\;{\geq}\;{\varepsilon}{\sqrt{2n\;{\log}\;{\log}\;n}}$) is replaced by E{$|S_n|/{\sqrt{n}}-{\varepsilon}{\sqrt{2\;{\log}\;{\log}\;n}$}+ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.

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References

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