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2009 Cramér Type Moderate deviations for the Maximum of Self-normalized Sums
Zhishui Hu, Qi-Man Shao, Qiying Wang
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Electron. J. Probab. 14: 1181-1197 (2009). DOI: 10.1214/EJP.v14-663

Abstract

Let $\{ X, X_i , i \geq 1\}$ be i.i.d. random variables, $S_k$ be the partial sum and $V_n^2 = \sum_{1\leq i\leq n} X_i^2$. Assume that $E(X)=0$ and $E(X^4) < \infty$. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that $P(\max_{1 \leq k \leq n} S_k \geq x V_n) / (1- \Phi(x)) \to 2$ uniformly in $x \in [0, o(n^{1/6}))$.

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Zhishui Hu. Qi-Man Shao. Qiying Wang. "Cramér Type Moderate deviations for the Maximum of Self-normalized Sums." Electron. J. Probab. 14 1181 - 1197, 2009. https://doi.org/10.1214/EJP.v14-663

Information

Accepted: 31 May 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1272.68116
MathSciNet: MR2511281
Digital Object Identifier: 10.1214/EJP.v14-663

Subjects:
Primary: 60F10
Secondary: 62E20

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