Precise asymptotics for complete 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\textbf{H},\|\cdot\|)$\end{document} with covariance operator Σ. Set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_n=\sum_{i=1}^nX_i,$\end{document}n ≥ 1. We prove that for 1 < p < 2 and r > 1 + p/2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{lll} &\lim\limits_{\varepsilon\searrow0}\varepsilon^{(2r-p-2)/(2-p)}\sum\limits_{n=1}^\infty n^{r/p-2-1/p}{\mbox{\rm{\textsf{E}}}}\{\|S_n\|-\sigma\varepsilon n^{1/p}\}_+\\&\quad\qquad\qquad\qquad=\sigma^{-(2r-2-p)/(2-p)}\frac{p(2-p)}{(r-p)(2r-p-2)}{\mbox{\rm{\textsf{E}}}}\|Y\|^{2(r-p)/(2-p)}, \end{array}$$\end{document}where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ2 is the largest eigenvalue of Σ.