A supplement to precise asymptotics in the law of the iterated logarithm

Let {X,Xn;n⩾1} be a sequence of real-valued i.i.d. random variables with E(X)=0 and E(X2)=1, and set Sn=∑i=1nXi, n⩾1. This paper studies the precise asymptotics in the law of the iterated logarithm. For example, using a result on convergence rates for probabilities of moderate deviations for {Sn;n⩾1} obtained by Li et al. [Internat. J. Math. Math. Sci. 15 (1992) 481-497], we prove that, for every b∈(−1/2,1], limɛ↓0ɛ(2b+1)/2∑n⩾3(loglogn)bnP(|Sn|⩾σn(2+ɛ)nloglogn+an)=e−2γ2b2/πΓ(b+(1/2)), whenever limn→∞(loglognn)1/2an=γ∈[−∞,∞], where Γ(s)=∫0∞ts−1e−tdt, s>0, σ2(t)=E(X2I(|X|