×

When is the Student \(t\)-statistic asymptotically standard normal? (English) Zbl 0958.60023

Summary: Let \(X\), \(X_i\), \(i\in\mathbb{N}\), be independent, identically distribued random variables. It is shown that the Student \(t\)-statistic based upon the sample \(\{X_i\}^n_{i=1}\) 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 \(U_n:= \sum^n_{i=1} X_i/ (\sum^n_{i=1} X_i^2)^{1/2}\), \(n\in\mathbb{N}\), are stochastically bounded, then they are uniformly sub-Gaussian, that is, \(\sup_n\mathbb{E} \exp(\lambda U^2_n) <\infty\) for some \(\lambda>0\).

MSC:

60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York. · Zbl 0457.60001
[2] Bentkus, V. and G ötze, F. (1994). The Berry-Esseen bound for Student’s statistic. Ann. Probab. 24 491-503. · Zbl 0855.62009
[3] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA. · Zbl 0174.48801
[4] Cs örg 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. · Zbl 0635.60022
[5] Cs örg 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. · Zbl 0657.60029
[6] Efron, B. (1969). Student’s t-test under symmetry conditions. J. Amer. Statist. Assoc. 64 1278- 1302. JSTOR: · Zbl 0188.50304
[7] 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. · Zbl 0214.17303
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. · Zbl 0219.60003
[9] Griffin, P. S. and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums. Proc. Cambridge Phil. Soc. 109 597-610. · Zbl 0723.62008
[10] Kahane, J.-P. (1968). Some Random Series of Functions. Heath, Lexington, MA. · Zbl 0192.53801
[11] Kwapie ń, S. and Woyczy ński, W. (1992). Random Series and Stochastic Integrals. Birkhäuser, Boston. · Zbl 0751.60035
[12] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632. · Zbl 0465.60031
[13] Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of selfnormalized sums. Ann. Probab. 1 788-809. · Zbl 0272.60016
[14] Maller, R. A. (1981). A theorem on products of random variables, with application to regression. Austral. J. Statist. 23 177-185. · Zbl 0483.60016
[15] O’Brien, G. L. (1980). A limit theorem for sample maxima and heavy branches in Galton-Watson trees. J. Appl. Probab. 17 539-545. JSTOR: · Zbl 0428.60034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.