| Precise Rates in the Generalized Law of the Iterated Logarithm in ${\mathbb{R}}^m$ |
| Received:February 06, 2017 Revised:August 04, 2017 |
| Key Words:
precise rates law of iterated logarithm complete convergence i.i.d. random vectors
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.61662037) and the Scientific Program of Department of Education of Jiangxi Province (Grant Nos.GJJ150894; GJJ150905). |
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| Abstract: |
| Let \{$X$, $X_n$, $n\ge 1$\} be a sequence of i.i.d. random vectors with ${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$ and ${\rm Cov}(X,X)=\sigma^2I_m$, and set $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. For every $d>0$ and $a_n=o((\log\log n)^{-d})$, the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of ${\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2018.01.010 |
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