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 , if and only if E[X21] < [infinity] and E[X1] = 0. We prove that there are absolute constants C1, C2 [set membership, variant] (0, [infinity]) such that if X1, X2, ... are independent identically distributed mean zero random variables, then c1[lambda]-2 E[X12·1{X1[lambda]}][less-than-or-equals, slant]S([lambda])[less-than-or-equals, slant]C2[lambda]-2 E[X12·1{X1[lambda]}], for every [lambda] > 0.