Asymptotics related to a series of T.L. Lai

Let {X,Xn,n≥1}{X,Xn,n≥1} be a sequence of i.i.d. random variables, and set Sn=X1+⋯+XnSn=X1+⋯+Xn. For 10δ>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↘1/2δ↘1/2.