The Annals of Probability

When is the Student $t$-statistic asymptotically standard normal?

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 $\lambda > 0$.

Article information

Source
Ann. Probab., Volume 25, Number 3 (1997), 1514-1531.

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404523

Digital Object Identifier
doi:10.1214/aop/1024404523

Mathematical Reviews number (MathSciNet)
MR1457629

Zentralblatt MATH identifier
0958.60023

Citation

Giné, Evarist; Götze, Friedrich; Mason, David M. When is the Student $t$-statistic asymptotically standard normal?. Ann. Probab. 25 (1997), no. 3, 1514--1531. doi:10.1214/aop/1024404523. https://projecteuclid.org/euclid.aop/1024404523

References

• Araujo, A. and Gin´e, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
• Bentkus, V. and G ¨otze, F. (1994). The Berry-Esseen bound for Student's statistic. Ann. Probab. 24 491-503.