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

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, and set Sn = X1 +· · ·+Xn, n 1. Let an = o( √ n/ log n). We prove that, for any 1 < r <3/2 and a >−d/2, lim ε √ r−1 ε2 − (r −1) a+d/2 ∞ n=1 nr−2(log n)aP Sn σφ(n)ε + an = Γ −1(d/2)K(Σ)(r − 1)(d−2)/2Γ (a +d/2) holds if EX = 0, E X 2r log X a−r <∞.  2004 Elsevier Inc. All rights reserved.