A converse to precise asymptotic results

Let {X,Xn,n⩾1} be i.i.d. random variables with partial sums {Sn,n⩾1}, put f(ε)=∑nanP(|Sn|⩾εbn),ε⩾0, and assume there exist functions g and h, such that limε↘0g(ε)f(ε)=h(EX2) whenever EX2<∞ and EX=0. We prove the converse result, namely that limsupε↘0g(ε)f(ε)<∞ and bn=O(n) imply EX2<∞ and EX=0.