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/logn). We prove that, for any 1−d/2, limɛ↘r−1[ɛ2−(r−1)]a+d/2∑n=1∞nr−2(logn)aP{‖Sn‖⩾σϕ(n)ɛ+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]<∞.