When is the Student $t$-statistic asymptotically standard normal?
Giné, Evarist ; Götze, Friedrich ; Mason, David M.
Ann. Probab., Tome 25 (1997) no. 4, p. 1514-1531 / Harvested from Project Euclid
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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 $\lambda > 0$.
Publié le : 1997-07-14
Classification:  Student $t$-statistic,  self-normalized sums,  domains of attraction,  convergence of moments,  60F05,  62E20 
@article{1024404523,
     author = {Gin\'e, Evarist and G\"otze, Friedrich and Mason, David M.},
     title = {When is the Student $t$-statistic asymptotically standard
 normal?},
     journal = {Ann. Probab.},
     volume = {25},
     number = {4},
     year = {1997},
     pages = { 1514-1531},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1024404523}
}

Giné, Evarist; Götze, Friedrich; Mason, David M. When is the Student $t$-statistic asymptotically standard
 normal?. Ann. Probab., Tome 25 (1997) no. 4, pp.  1514-1531. http://gdmltest.u-ga.fr/item/1024404523/