ON THE RATES OF THE ALMOST SURE CONVERGENCE FOR SELF-NORMALIZED LAW OF THE ITERATED LOGARITHM

Title & Authors
ON THE RATES OF THE ALMOST SURE CONVERGENCE FOR SELF-NORMALIZED LAW OF THE ITERATED LOGARITHM
Pang, Tian-Xiao;

Abstract
Let {$\small{X_i}$, $\small{i{\geq}1}$} be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote $\small{S_n={\sum}_{i=1}^n\;X_i}$, $\small{M_n=max_{1{\leq}i{\leq}n}\;{\mid}S_i{\mid}}$ and $\small{V_n^2={\sum}_{i=1}^n\;X_i^2}$. Then for d > -1, we showed that under some regularity conditions, $\small{\lim_{{\varepsilon}{\searrow}0}{\varepsilon}^2^{d+1}\sum_{n=1}^{\infty}\frac{(loglogn)^d}{nlogn}I\{M_n/V_n{\geq}\sqrt{2loglogn}({\varepsilon}+{\alpha}_n)\}=\frac{2}{\sqrt{\pi}(1+d)}{\Gamma}(d+3/2)\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^{2d+2}}\;a.s.}$ holds in this paper, where If g denotes the indicator function.
Keywords
almost sure convergence;self-normalized;domain of attraction of the normal law;law of the iterated logarithm;
Language
English
Cited by
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