In the paper, the complete moment convergence is obtained for i.i.d. random variables such that all moments exist, but the moment generating function does not exist. The main results extend the related known works due to Gut and Stadtmüller.

For a sequence of i.i.d. random variables {X,Xn,n≥1} with EX=0 and Eexp{(log|X|)α}<∞ for some α>1, Gut and Stadtmüller (2011) proved a Baum–Katz theorem. In this paper, it is proved that 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).

If Y1,Y2,… is a sequence of random variables such that Yn⟶a.s.Y as n→∞, and {τ(t),t≥0} is a family of “indices” such that τ(t)⟶a.s.∞ as t→∞, then it is pretty obvious that Yτ(t)⟶a.s.Y as t→∞. However, if one relaxes one of ⟶a.s. to ⟶p and lets the other one...

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables, and set Sn=X1+⋯+Xn. For 1<p≤3/2, if EX=0, EX2=1 and E[|X|2p(log+|X|)−p]<∞, we prove that limε↘0ε1/2∑n≥2np−2P(|Sn|≥(2p−2+ε)nlogn)=2p−1, thus extending a result by Spătaru (2004). For 1<p≤3/2 and δ>0, we establish conditions for the convergence of the related series ∑n≥2np−2P(|Sn|≥(2p−2)nlogn+δn1/2loglogn(2p−2)nlogn), and derive its precise asymptotic as δ↘1/2.