Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let {X, Xn; n ≥ 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with covariance operator Σ. Set Sn = X1 + X2 + ... + Xn, n ≥ 1. We prove that, for b > −1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(logn)^b }} {{n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }}} E\{ \left\| {S_n } \right\| - \sigma \varepsilon \sqrt {nlogn} \} _ + = \frac{{\sigma ^{ - 2(b + 1)} }} {{^{(2b + 3)(b + 1)} }}E\left\| Y \right\|^{2b + 3} $$\end{document} holds if EX = 0, and E‖X‖2(log ‖X‖)3b∨(b+4) < ∞, where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ2 denotes the largest eigenvalue of Σ.