Sepanski, Steven J. Asymptotics for multivariate \(t\)-statistic for random vectors in the generalized domain of attraction of the multivariate normal law. (English) Zbl 0862.62016 Stat. Probab. Lett. 30, No. 2, 179-188 (1996). Summary: We define the appropriate analogue of Student’s \(t\)-statistic for multivariate data, and prove that it is asymptotically normal for random vectors in the Generalized Domain of Attraction of the Normal Law. This extends an earlier result of the author [J. Multivariate Anal. 49, No. 1, 41-54 (1994; Zbl 0796.62019)] where asymptotic normality was proved under a stronger hypothesis on the Domain of Attraction. Cited in 6 Documents MSC: 62E20 Asymptotic distribution theory in statistics 62F12 Asymptotic properties of parametric estimators 60F05 Central limit and other weak theorems 62H12 Estimation in multivariate analysis Keywords:Student’s \(t\)-statistic; generalized domain of attraction; affine normalization; self normalization; bootstrap; central limit theorem; asymptotic normality Citations:Zbl 0796.62019 PDFBibTeX XMLCite \textit{S. J. Sepanski}, Stat. Probab. Lett. 30, No. 2, 179--188 (1996; Zbl 0862.62016) Full Text: DOI References: [1] Araujo, A.; Giné, E., The Central Limit Theorem for Real and Banach Valued Random Variables (1980), Wiley: Wiley New York [2] Billingsley, P., Convergence of types in \(k\)-space, Z. Wahrsch. Verw. Gebiete., 5, 175-179 (1966) · Zbl 0152.17102 [3] Feller, W., (An Introduction to Probability Theory and Its Applications, Vol. 2 (1971), Wiley: Wiley New York) · Zbl 0219.60003 [4] Griffin, P. S.; Mason, D. M., On the asymptotic normality of self-normalized sums, (Math. Proc. Camb. Phil. Soc., 109 (1991)), 597-610 · Zbl 0723.62008 [5] Hahn, M. G., The generalized domain of attraction of a Gaussian law on Hilbert space, (Proc. 2nd Conf. on Probability in Banach Spaces. Proc. 2nd Conf. on Probability in Banach Spaces, Lecture Notes in Math., Vol. 709 (1979)), 125-144 [6] Hahn, M. G.; Klass, M. J., Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal, Ann. Probab., 8, 262-280 (1980) · Zbl 0428.60032 [7] Hahn, M. G.; Klass, M. J., The generalized domain of attraction of spherically symmetric stable laws on \(R^d\), (Proc. Conf. on Probability in Vector Spaces II. Proc. Conf. on Probability in Vector Spaces II, Poland. Proc. Conf. on Probability in Vector Spaces II. Proc. Conf. on Probability in Vector Spaces II, Poland, Lecture Notes in Math., Vol. 828 (1980)), 52-81 [8] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0576.15001 [9] Maller, R. A., Quadratic negligibility and the asymptotic normality of operator normed sums, J. Multivariate Anal., 44, 191-219 (1993) · Zbl 0773.60016 [10] Sepanski, S. J., Asymptotic normality of multivariate t and Hotelling’s \(T^2\) statistics under infinite second moments via bootstrapping, J. Mutlivariate Anal., 49, 41-54 (1994) · Zbl 0796.62019 [11] Sepanski, S. J., Necessary and sufficient conditions for the multivariate bootstrap of the mean, Statist. Probab. Lett., 19, 205-216 (1994) · Zbl 0796.62020 [12] Sepanski, S. J., Probabilistic characterizations of the generalized domain of attraction of the multivariate normal law, J. Theoret. Probab., 7, 857-866 (1994) · Zbl 0807.60033 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.