Convergence rate in precise asymptotics for Davis law of large numbers

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with E[X]=0 and E[X2]=σ2∈(0,∞), and set Sn=∑k=1nXk,n≥1. For any δ≥0, let γδ=limn→∞(∑j=1n(logj)δj−(logn)δ+1δ+1)andηδ=∑n=1∞(logn)δnP(Sn=0). Under the moment condition E[X2(log(1+∣X∣))1+δ]<∞, we prove that limϵ↘0[∑n=1∞(logn)δnP(∣Sn∣≥ϵnlogn)−E[∣N∣2δ+2]δ+1σ2δ+2ϵ−(2δ+2)]=γδ−ηδ, which refines Theorem 3 of Gut and Spătaru (2000a).