Electronic Journal of Probability

Limit Theorems for Self-Normalized Large Deviation

Qiying Wang

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Let $X, X_1, X_2, \cdots $ be i.i.d. random variables with zero mean and finite variance $\sigma^2$. It is well known that a finite exponential moment assumption is necessary to study limit theorems for large deviation for the standardized partial sums. In this paper, limit theorems for large deviation for self-normalized sums are derived only under finite moment conditions. In particular, we show that, if $EX^4 < \infty$, then $$ \frac{P(S_n /V_n \geq x)}{1-\Phi(x)} \exp\left\{ -\frac{x^3 EX^3}{3\sqrt{ n}\sigma^3} \right\} \left[ 1 + O\left(\frac{1+x} {\sqrt {n}}\right) \right], $$ for $x \ge 0$and $x = O(n^{1/6})$, where $S_n\sum_{i=1}^nX_i$ and $V_n (\sum_{i=1}^n X_i^2)^{1/2}$.

Article information

Electron. J. Probab., Volume 10 (2005), paper no. 38, 1260-1285.

Accepted: 14 November 2005
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory

Cram'er large deviation limit theorem

This work is licensed under aCreative Commons Attribution 3.0 License.


Wang, Qiying. Limit Theorems for Self-Normalized Large Deviation. Electron. J. Probab. 10 (2005), paper no. 38, 1260--1285. doi:10.1214/EJP.v10-289. https://projecteuclid.org/euclid.ejp/1464816839

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