Precise asymptotics in complete moment convergence for self-normalized sums

Let X,X1,X2,…X,X1,X2,… be a sequence of nondegenerate i.i.d. random variables with zero means, set Sn=X1+⋯+XnSn=X1+⋯+Xn and Vn2=X12+⋯+Xn2, EX2I(|X|≤x)EX2I(|X|≤x) is a slowly varying function at ∞∞. We prove that, for any β>2,δ>2/β−1β>2,δ>2/β−1,limϵ↓0ϵβ(δ+1)−2∑n=2∞(logn)δ−2/βnE(Sn/Vn)2I(|Sn|≥ϵVn(logn)1/β)=βE|N|β(δ+1)β(δ+1)−2, where NN is a standard normal random variable.