Precise Rates in the Generalized Law of the Iterated Logarithm in the Hilbert Space

Let \X,X_n,n\ge1\ be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (\mathbbH,\|\cdot\|) with covariance operator \Sigma, set S_n=X_1+X_2+\cdots+X_n, n\ge 1. For every m>0 and a_n=O((\ln\ln n)^-2m), we study the precise rates in the generalized law of the iterated logarithm for a kind of weighted infinite series of \pr\\|S_n\|\ge(\epsilon+a_n)\sigma\sqrtn(\ln\ln n)^m\. Let \beta_n(\epsilon)=o(\sqrt1/\ln\ln n). We also prove that, for any r>1 and \alpha>-d/2, \beginalign* &\lim_\epsilon\searrow\sqrtr-1\epsilon^2-(r-1)^\alpha+d/2 \tsm_n=1^\infty\frac1n(\ln n)^r-2(\ln\ln n)^\alpha \pr\\|S_n\|\ge\sigma\psi(n)\epsilon+\beta_n(\epsilon)\\\ =\;&\Gamma^-1(d/2)K(\Sigma)(r-1)^(d-2)/2\Gamma(\alpha+d/2) \endalign* holds if \ep X=0, \ep\|X\|^2(\ln\|X\|)^r-1<\infty.