A supplement to the Davis–Gut law

Let {X,Xn;n⩾1}{X,Xn;n⩾1} be a sequence of i.i.d. real-valued random variables and set Sn=∑i=1nXi, n⩾1n⩾1. Let h(⋅)h(⋅) be a positive nondecreasing function such that ∫1∞dtth(t)=∞. Define Lt=logemax{e,t}Lt=logemax{e,t} for t⩾0t⩾0. In this note we prove that∑n=1∞1nh(n)P(|Sn|⩾(1+ε)2nLψ(n)){<∞,ifε>0,=∞,ifε<0 if and only if E(X)=0E(X)=0 and E(X2)=1E(X2)=1, where ψ(t)=∫1tdssh(s), t⩾1t⩾1. When h(t)≡1h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=r(Lt)h(t)=(Lt)r, 0