A supplement to the Davis–Gut law

Let {X,Xn;n 1} be a sequence of i.i.d. real-valued random variables and set Sn = n i=1 Xi , n 1. Let h(·) be a positive nondecreasing function such that ∞1 dt th(t) =∞. Define Lt = loge max{e, t} for t 0. In this note we prove that ∞ n=1 1 nh(n) P |Sn| (1+ ε) 2nLψ(n) <∞, if ε >0, =∞, if ε <0 if and only if E(X) = 0 and E(X2) = 1, where ψ(t) = t 1 ds sh(s) , t 1. When h(t) ≡ 1, this result yields what is called the Davis–Gut law. Specializing our result to h(t) = (Lt)r, 0 < r 1, we obtain an analog of the Davis–Gut law. © 2006 Elsevier Inc. All rights reserved.