A two-sided estimate in the Hsu—Robbins—Erdös law of large numbers

Let X1, X2, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that S(λ)def∑n=1∞P(|X1+⋯+Xn|⩾λn)<∞, ∀λ>0, 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}], ery λ > 0.