Cram'er Type Moderate deviations for the Maximum of Self-normalized Sums
Zhishui Hu, USTC
Qi-Man Shao, HKUST
Qiying Wang, University of Sydney
Abstract
Let { X, Xi , i ≥ 1} be i.i.d. random variables, Sk be the partial sum and Vn2
= ∑1&le i&le n Xi2. Assume that E(X)=0 and E(X4) < ∞. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that P(max1 ≤ k ≤ n Sk &ge x Vn) / (1- Φ(x)) → 2 uniformly in x ∈ [0, o(n1/6)).
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