Let \{X,X_n,n\ge1\} be a sequence of i.i.d. random vectors with \ep X=(0,0,\cdots,0)_{m\times1} and \cov(X,X)=\Sigma, and set S_n=\tsm_{i=1}^nX_i, n\ge1. For every d>0 and a_n=o((\ln\ln n)^{-d}), we obtain the precise rates in the generalized law of the iterated logarithm for a kind of weighted infiniteseries of \pr(|S_n|\ge(\varepsilon+a_n)\sigma\sqrt{n}(\ln\ln n)^d).