A Supplement to Precise Asymptotics in the Law of the Iterated Logarithm for Self-normalized Sums

Let X, X1, X2, . . . be i.i.d. random variables with zero means, variance one, and set Sn = ∑n i=1 Xi, n ≥ 1. Gut and Spǎtaru [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P(|Sn| ≥ ε √ 2n log log n) is replaced by E{|Sn|/√n− ε √ 2 log log n}+ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.