Rate of convergence in a theorem of Heyde

Let {X,Xn,n≥1}{X,Xn,n≥1} be a sequence of i.i.d. random variables with mean zero, and set Sn=∑k=1nXk, TX(t)=EX2I(|X|>t)TX(t)=EX2I(|X|>t). Heyde (1975) proved precise asymptotics for ∑n=1∞P(|Sn|≥nϵ) as ϵ↘0ϵ↘0. In this paper, we obtain a convergence rate in a theorem of Heyde (1975) under a second moment assumption only. Furthermore, under the additional assumption of TX(t)=O(t−δ)TX(t)=O(t−δ) as t→∞t→∞ for some δ>0δ>0, we obtain a refined result.