## Probability Surveys

### Self-normalized limit theorems: A survey

#### Abstract

Let $X_{1},X_{2},\ldots,$ be independent random variables with $EX_{i}=0$ and write $S_{n}=\sum_{i=1}^{n}X_{i}$ and $V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}$. This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum $S_{n}/V_{n}$. Other self-normalized limit theorems are also briefly discussed.

#### Article information

Source
Probab. Surveys, Volume 10 (2013), 69-93.

Dates
First available in Project Euclid: 28 November 2013

https://projecteuclid.org/euclid.ps/1385665279

Digital Object Identifier
doi:10.1214/13-PS216

Mathematical Reviews number (MathSciNet)
MR3161676

Zentralblatt MATH identifier
1286.60029

#### Citation

Shao, Qi-Man; Wang, Qiying. Self-normalized limit theorems: A survey. Probab. Surveys 10 (2013), 69--93. doi:10.1214/13-PS216. https://projecteuclid.org/euclid.ps/1385665279

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