Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let {X,Xn; n 1} be a sequence of i.i.d. random variables with EX = 0 and EX2 = σ2 < ∞. Set Sn = n k=1 Xk, Mn = maxk n |Sk|, n 1. Let r >1, then we obtain lim ε √r−1 1 −log(ε2 − (r −1)) ∞ n=1 nr−2−1/2E Mn −σε 2n log n + = 2σ (r −1)√2π holds, if and only if EX = 0, EX2 = σ2 <∞and E(|X|2r /(log |X|)r) <∞. © 2006 Elsevier Inc. All rights reserved.