Precise rates in the generalized law of the iterated logarithm in $\rr^m$
Precise rates in the generalized law of the iterated logarithm in $\rr^m$

DOI：

 作者 单位 E-mail 徐明周 景德镇陶瓷大学 mingzhouxu@whu.edu.cn 丁云正 景德镇陶瓷大学 周永正 景德镇陶瓷大学

Let \{$X$, $X_n$, $n\ge 1$\} be a sequence of i.i.d. random vectors with $\ee X=(0,\cdots,0)_{m\times 1}$ and $\cov(X,X)=\sigma^2I_m$, 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 $\pp(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$.