Abstract :
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.
