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

Abstract

Let { X , X n ; n ⩾ 1 } be a sequence of real-valued i.i.d. random variables with E ( X ) = 0 and E ( X 2 ) = 1 , and set S n = ∑ i = 1 n X i , 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 { S n ; 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 ɛ ↓ 0 ɛ ( 2 b + 1 ) / 2 ∑ n ⩾ 3 ( log log n ) b n P ( | S n | ⩾ σ n ( 2 + ɛ ) n log log n + a n ) = e − 2 γ 2 b 2 / π Γ ( b + ( 1 / 2 ) ) , whenever lim n → ∞ ( log log n n ) 1 / 2 a n = γ ∈ [ − ∞ , ∞ ] , where Γ ( s ) = ∫ 0 ∞ t s − 1 e − t d t , s > 0 , σ 2 ( t ) = E ( X 2 I ( | X | < t ) ) − ( E ( X I ( | X | < t ) ) ) 2 , t ⩾ 0 , and σ n 2 = 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 Spătaru [Ann. Probab. 28 (2000) 1870–1883].