Abstract: | Let X1, X2, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that , if and only if E[X21] < ∞ and E[X1] = 0. We prove that there are absolute constants C1, C2 (0, ∞) such that if X1, X2, … are independent identically distributed mean zero random variables, then c1λ−2 E[X12·1{|X1|λ}]S(λ)C2λ−2 E[X12·1{|X1|λ}] ,for every λ > 0. |