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

Abstract. 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 < 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−22 Γ(a+ d/2) holds if EX = 0, E[(X, y)]2 <∞, E[‖X‖2r(log ‖X‖)a−r] <∞, and E[(X, ei)2I{|(X, ei) |> t}] = o ( 1log t), as t→∞, for ∀i