A supplement to the complete convergence

Let X,X1,X2,…X,X1,X2,… be independent identically distributed random variables. Then, Baum and Katz [1965. Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123] proved that for p>1p>1,∑k=1∞kp-2P(|Sk|⩾k)<∞,if and only if E|X|p<∞E|X|p<∞ and EX=0EX=0. We prove thatf(n)≔∑k=1nkp-2P(|Sk|⩾k)is slowly varying asn→∞,if and only if EX=0EX=0 and E|X|pI{|X|⩽x}E|X|pI{|X|⩽x} is slowly varying as x→∞x→∞.