An estimate of the remainder of a limit theorem

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with zero mean and finite variance. Set Sn=∑k=1nXk, EX2=σ2>0, λα(ϵ)=∑n=1∞P(|Sn|≥n1/2+αϵ), 0<α<1. In this paper, we discuss the rate of the approximation of σ1/αcα by ϵ1/αλα(ϵ) under suitable conditions, and extend the results of Klesov (1994), and He and Xie (in press), where cα=π−1/221/2αΓ(12+12α).