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- Title
Exact Rates in the Davis-Gut Law of Iterated Logarithm for the First-Moment Convergence of Independent Identically Distributed Random Variables.
- Authors
Xiao, X.-Y.; Yin, H.-W.
- Abstract
Let { X, X , n ≥ 1} be a sequence of independent identically distributed random variables and let $$ {S}_n={\sum}_{i=1}^n{X}_i $$ , M = max ≤ k ≤ n| S |. For r > 0 , let a ( ε) be a function of ε such that a ( ε) log log n→ τ as n→∞ and $$ \varepsilon \searrow \sqrt{r} $$ . If $$ \mathbb{E}{X}^2I\left\{\left|X\right|\ge t\right\}=o\left({\left(\log\;\log\;t\right)}^{-1}\right) $$ as t→∞, then, by using the strong approximation, we show that holds if and only if $$ \mathbb{E}X=0 $$ , $$ \mathbb{E}{X}^2={\sigma}^2 $$ , and $$ \mathbb{E}{X}^2{\left(\log \left|X\right|\right)}^{r-1}{\left(\log\;\log\;\left|X\right|\right)}^{-\frac{1}{2}}<\infty $$ .
- Subjects
RANDOM variables; ITERATIVE methods (Mathematics); STOCHASTIC convergence; MATHEMATICAL sequences; WIENER processes
- Publication
Ukrainian Mathematical Journal, 2017, Vol 69, Issue 2, p283
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-017-1361-3