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

Let {X, X n ; n . 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H, . · .) with covariance operator .. Set S n = X 1 + X 2 + ... + X n , n . 1. We prove that, for b > .1, lim..0.2(b+1).n=1.(logn)bn3/2E{.Sn....nlogn......}+=..2(b+1)(2b+3)( b+1)E.Y.2b+3 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 ., and . 2 denotes the largest eigenvalue of ..