Precise rates in the law of the logarithm in the Hilbert space

Let {X, X-n; n >= 1} be a sequence of i.i.d. randorn variables taking values in a real separable Hilbert space (H, parallel to (.) parallel to) with covariance operator Sigma, and set S-n = X-l + (. . .) + X-n, n > 1. Let a(n) = o(root n/log n). We prove that, for any 1 < r < 3/2 and a > -d/2, lim(epsilon SE arrow root r-1) [epsilon(2) - (r - 1)](a+d/2) Sigma(infinity)(n=1)n(r-2)(log n)P-a{parallel to Sn parallel to >= sigma phi (n)epsilon + a(n)} =Gamma(-1)(d/2) K (Sigma) (r - 1)((d-2)/2) Gamma(a + d/2) holds if EX = 0, E[ parallel to X parallel to(2r) (log parallel to X parallel to)(a-4)] < infinity. (c) 2004 Elsevier Inc. All rights reserved.