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July 1997 When is the Student $t$-statistic asymptotically standard normal?
Evarist Giné, Friedrich Götze, David M. Mason
Ann. Probab. 25(3): 1514-1531 (July 1997). DOI: 10.1214/aop/1024404523

Abstract

Let $X, X_i, i \in \mathbb{N}$, be independent, identically distributed random variables. It is shown that the Student $t$-statistic based upon the sample ${X_i}_{i=1}^n$ is asymptotically $N(0, 1)$ if and only if $X$ is in the domain of attraction of the normal law. It is also shown that, for any $X$, if the self-normalized sums $U_n := \sum_{i=1}^n X_i/(\sum_{i=1}^n X_i^2)^{1/2}, n \in \mathbb{N}$, are stochastically bounded then they are uniformly subgaussian that is, $\sup_n \mathbb{E} \exp (\lambda U_n^2) < \infty$ for some $\lambda > 0$.

Citation

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Evarist Giné. Friedrich Götze. David M. Mason. "When is the Student $t$-statistic asymptotically standard normal?." Ann. Probab. 25 (3) 1514 - 1531, July 1997. https://doi.org/10.1214/aop/1024404523

Information

Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0958.60023
MathSciNet: MR1457629
Digital Object Identifier: 10.1214/aop/1024404523

Subjects:
Primary: 60F05, 62E20

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 3 • July 1997
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