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Central Limit Theorem and Near classical Berry-Esseen rate for self normalized sums in high dimensions

Abstract

In this article, we are interested in the high dimensional normal approximation of Tn=(i=1nXi1/(i=1nXi12),,T_n =\Big(\sum_{i=1}^{n}X_{i1}/\Big(\sqrt{\sum_{i=1}^{n}X_{i1}^2}\Big),\dots, i=1nXip/(i=1nXip2))\sum_{i=1}^{n}X_{ip}/\Big(\sqrt{\sum_{i=1}^{n}X_{ip}^2}\Big)\Big) in Rp\mathcal{R}^p uniformly over the class of hyper-rectangles Are={j=1p[aj,bj]R:ajbj,j=1,,p}\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leq b_j \leq \infty, j=1,\ldots,p\}, where X1,,XnX_1,\dots,X_n are non-degenerate independent pp-dimensional random vectors. We assume that the components of XiX_i are independent and identically distributed (iid) and investigate the optimal cut-off rate of logp\log p in the uniform central limit theorem (UCLT) for TnT_n over Are\mathcal{A}^{re}. The aim is to reduce the exponential moment conditions, generally assumed for exponential growth of the dimension with respect to the sample size in high dimensional CLT, to some polynomial moment conditions. Indeed, we establish that only the existence of some polynomial moment of order [2,4]\in [2,4] is sufficient for exponential growth of pp. However the rate of growth of logp\log p can not further be improved from o(n1/2)o\big(n^{1/2}\big) as a power of nn even if XijX_{ij}'s are iid across (i,j)(i,j) and X11X_{11} is bounded. We also establish nearnκ/2-n^{-\kappa/2} Berry-Esseen rate for TnT_n in high dimension under the existence of (2+κ)(2+\kappa)th absolute moments of XijX_{ij} for 0<κ10< \kappa \leq 1. When κ=1\kappa =1, the obtained Berry-Esseen rate is also shown to be optimal. As an application, we find respective versions for component-wise Student's t-statistic, which may be useful in high dimensional statistical inference.

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