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2013 Self-normalized limit theorems: A survey
Qi-Man Shao, Qiying Wang
Probab. Surveys 10: 69-93 (2013). DOI: 10.1214/13-PS216


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.


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Qi-Man Shao. Qiying Wang. "Self-normalized limit theorems: A survey." Probab. Surveys 10 69 - 93, 2013.


Published: 2013
First available in Project Euclid: 28 November 2013

zbMATH: 1286.60029
MathSciNet: MR3161676
Digital Object Identifier: 10.1214/13-PS216

Primary: 60F05 , 60F17
Secondary: 62E20

Keywords: absolute error , central limit theorem , convergence rate , Cramér moderate deviation , Darling-Erdös theorem , Hotelling’s $T^{2}$ statistic , invariance principle , large deviation , Laws of the iterated logarithm , relative error , saddle-point approximation , self-normalized sum , Student $t$ statistic

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.10 • 2013
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