Precise asymptotics for a new kind of complete moment convergence

In this paper, we introduce a new kind of complete moment convergence, which includes complete convergence as a special case. And we also achieves some results about precise asymptotics for this kind of complete moment convergence. For example, let {X,Xn;n⩾1}{X,Xn;n⩾1} be a sequence of i.i.d. random variables, then limε↘01-logε∑n=1∞1n2ESn2I{|Sn|⩾εn}=2σ2holds, if and only if EX=0EX=0, EX2=σ2EX2=σ2 and EX2log+|X|<∞.