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 Probability Surveys > Vol. 10 (2013) open journal systems 


Self-normalized limit theorems: A survey

Qi-Man Shao, Chinese University of Hong Kong
Qiying Wang, University of Sydney


Abstract
Let \(X_1, X_2, \ldots,\) be independent random variables with \(EX_i=0\) and write \(S_n=\sum_{i=1}^nX_i\) and \(V_n^2=\sum_{i=1}^nX_i^2\). 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.

AMS 2000 subject classifications: Primary 60F05, 60F17; secondary 62E20.

Keywords: Self-normalized sum, Student t statistic, central limit theorem, invariance principle, convergence rate, absolute error, relative error, Cramér moderate deviation, large deviation, saddle-point approximation, laws of the iterated logarithm, Darling-Erdös theorem, Hotelling’s T2 statistic.

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

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