On the moment convergence rates of LIL in Hilbert space

Let {X,Xn;n≥1}{X,Xn;n≥1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H,‖⋅‖) with mean zero and covariance ΣΣ and set Sn=∑k=1nXk,n≥1. Let σi2,i≥1 be the eigenvalues of ΣΣ arranged in non-increasing order and take into account the multiplicities. Let ll be the dimension of the corresponding eigenspace of largest eigenvalue σ2=σ12. Let logx=ln(x∨e),x≥0logx=ln(x∨e),x≥0. This paper studies the precise rates for ∑n=1∞(logn)bn3/2E{‖Sn‖−σε2nloglogn}+. We show that when l>1l>1 and b>−1,E(‖X‖2(log‖X‖)3b+3/loglog‖X‖)<∞ implies limε↘1+b(1−1+bε2)ł−12∑n=1∞(logn)bn3/2E{‖Sn‖−σε2nloglogn}+=12(b+1)Γ−1(l/2)K′(Σ)Γ((l−1)/2), where Γ(⋅) is the Gamma function and K′(Σ)=∏i=l+1∞((σ2−σi2)/σ4)−1/2.