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October 2003 Self-normalized Cramér-type large deviations for independent random variables
Bing-Yi Jing, Qi-Man Shao, Qiying Wang
Ann. Probab. 31(4): 2167-2215 (October 2003). DOI: 10.1214/aop/1068646382


Let $X_1, X_2, \ldots $ be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramér-type large deviation result for the standardized partial sums. In this paper, we show that a Cramér-type large deviation theorem holds for self-normalized sums only under a finite $(2+\delta)$th moment, $0< \delta \leq 1$. In particular, we show $P(S_n /V_n \geq x)=\break (1-\Phi(x)) (1+O(1) (1+x)^{2+\delta} /d_{n,\delta}^{2+\delta})$ for $0 \leq x \leq d_{n,\delta}$,\vspace{1pt} where $d_{n,\delta} = (\sum_{i=1}^n EX_i^2)^{1/2}/(\sum_{i=1}^n E|X_i|^{2+\delta})^{1/(2+\delta)}$ and $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.


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Bing-Yi Jing. Qi-Man Shao. Qiying Wang. "Self-normalized Cramér-type large deviations for independent random variables." Ann. Probab. 31 (4) 2167 - 2215, October 2003.


Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1051.60031
MathSciNet: MR2016616
Digital Object Identifier: 10.1214/aop/1068646382

Primary: 60F10 , 60F15
Secondary: 60G50 , 62F03

Keywords: large deviation , Law of the iterated logarithm , Moderate deviation , nonuniform Berry--Esseen bound , self-normalized sum , Studentized bootstrap , t-statistic

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • October 2003
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