A SUPPLEMENT TO PRECISE ASYMPTOTICS IN THE LAW OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED SUMS

Title & Authors
A SUPPLEMENT TO PRECISE ASYMPTOTICS IN THE LAW OF THE ITERATED LOGARITHM FOR SELF-NORMALIZED SUMS
Hwang, Kyo-Shin;

Abstract
Let X, $\small{X_1}$, $\small{X_2}$, ... be i.i.d. random variables with zero means, variance one, and set $\small{S_n\;=\;{\sum}^n_{i=1}\;X_i}$, $\small{n\;{\geq}\;1}$. Gut and $\small{Sp{\check{a}}taru}$ [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($\small{|S_n|\;{\geq}\;{\varepsilon}{\sqrt{2n\;{\log}\;{\log}\;n}}}$) is replaced by E{$\small{|S_n|/{\sqrt{n}}-{\varepsilon}{\sqrt{2\;{\log}\;{\log}\;n}}$}+ 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.
Keywords
precise asymptotics;law of iterated logarithm;self-normalized sums;
Language
English
Cited by
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