A Baum–Katz theorem for i.i.d. random variables with higher order moments

For a sequence of i.i.d. random variables {X,Xn,n≥1}{X,Xn,n≥1} with EX=0EX=0 and Eexp{(log|X|)α}<∞Eexp{(log|X|)α}<∞ for some α>1α>1, Gut and Stadtmüller (2011) proved a Baum–Katz theorem. In this paper, it is proved that Eexp{(log|X|)α}<∞Eexp{(log|X|)α}<∞ if and only if ∑n=1∞exp{(logn)α}n−2(logn)α−1P(|Sn|>n)<∞, where Sn=∑i=1nXi. This result improves the corresponding one of Gut and Stadtmüller (2011).