Self-normalized limit theorems: a survey. (English) Zbl 1286.60029

Summary: Let \(X_{1},X_{2},\dots,\) be independent random variables with \(\operatorname{E}X_{i}=0\) 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.


60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
62E20 Asymptotic distribution theory in statistics
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