Precise asymptotics in complete moment convergence for self-normalized sums

Let X,X1,X2,… be a sequence of nondegenerate i.i.d. random variables with zero means, set Sn=X1+ +Xn and , EX2I(X≤x) is a slowly varying function at ∞. We prove that, for any β>2,δ>2/β−1, where N is a standard normal random variable.