## The Annals of Probability

### Self-normalized Cramér-type large deviations for independent random variables

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 2167-2215.

Dates
First available in Project Euclid: 12 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1068646382

Digital Object Identifier
doi:10.1214/aop/1068646382

Mathematical Reviews number (MathSciNet)
MR2016616

Zentralblatt MATH identifier
1051.60031

#### Citation

Jing, Bing-Yi; Shao, Qi-Man; Wang, Qiying. Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 (2003), no. 4, 2167--2215. doi:10.1214/aop/1068646382. https://projecteuclid.org/euclid.aop/1068646382

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