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 = n i=1 Xi , 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], lim ε↓0 ε(2b+1)/2 n 3 (log log n)b n P |Sn| σn (2 + ε)nlog log n+ an = e − √ 2 γ 2b 2/πΓ b + (1/2) , whenever limn→∞ log logn n 1/2 an = γ ∈ [−∞,∞], where Γ (s) = ∞ 0 t s−1e −t dt, s >0, σ 2(t )= E(X2I (|X| < √ t )) − (E(XI (|X| < √ t )))2, t 0, and σ 2 n = σ 2(n log log n), n 3. This result generalizes and improves Theorem 2.8 of Li et al. [Internat. J. Math. Math. Sci. 15 (1992) 481–497] and Theorem 1 of Gut and Sp˘ataru [Ann. Probab. 28 (2000) 1870–1883].  2004 Elsevier Inc. All rights reserved.