Asymptotics for Multivariate t-Statistic and Hotelling′s T2-Statistic Under Infinite Second Moments via Bootstrapping

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 domain of attraction of the normal law. We also prove that Hotelling′s T2-statistic has a chi-squared limiting distribution for random vectors in the generalized domain of attraction of the normal law. Our tool in proving these results is the bootstrap. We prove that the bootstrap version of the multivariate t-statistic is asymptotically normal when the parent distribution is in the generalized domain of attraction of the normal law.