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Cram'er Type Moderate deviations for the Maximum of Self-normalized Sums

   
@article{EJP663,
	author = {Zhishui Hu and Qi-Man Shao and Qiying Wang},
	title = {Cram'er Type Moderate deviations for the Maximum of Self-normalized Sums},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {14},
	year = {2009},
	keywords = {Large deviation, moderate deviation, self-normalized maximal sum},
	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 $x \in [0, o(n^{1/6}))$.},
	pages = {no. 41, 1181-1197},
	issn = {1083-6489},
	doi = {10.1214/EJP.v14-663},    
        url = {http://ejp.ejpecp.org/article/view/663}}