Precise Rates in the Generalized Law of the Iterated Logarithm in ${\mathbb{R}}^m$

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 ${\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$.