The Annals of Probability

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

Evarist Giné, Friedrich Götze, and David M. Mason

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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 $\lambda > 0$.

Article information

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

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

Student $t$-statistic self-normalized sums domains of attraction convergence of moments


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.

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  • 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.
  • Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.
  • Cs ¨org o, S. (1989). Notes on extreme and self-normalised sums from the domain of attraction of a stable law. J. London Math. Soc. 39 369-384.
  • Cs ¨org o, S., Haeusler, E. and Mason, D. M. (1988). A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. in Appl. Math. 9 259-333.
  • Efron, B. (1969). Student's t-test under symmetry conditions. J. Amer. Statist. Assoc. 64 1278- 1302.
  • Feller, W. (1966). On regular variation and local limit theorems. In Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 373-388. Univ. California Press, Berkeley.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • Griffin, P. S. and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums. Proc. Cambridge Phil. Soc. 109 597-610.
  • Kahane, J.-P. (1968). Some Random Series of Functions. Heath, Lexington, MA.
  • Kwapie ´n, S. and Woyczy ´nski, W. (1992). Random Series and Stochastic Integrals. Birkh¨auser, Boston.
  • LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632.
  • Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of selfnormalized sums. Ann. Probab. 1 788-809.
  • Maller, R. A. (1981). A theorem on products of random variables, with application to regression. Austral. J. Statist. 23 177-185.
  • O'Brien, G. L. (1980). A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Probab. 17 539-545.