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2009 Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors
Larry Goldstein, Qi-Man Shao
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Electron. Commun. Probab. 14: 474-485 (2009). DOI: 10.1214/ECP.v14-1502

Abstract

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized projection $W_\theta=Y_\theta/v_\theta$ of ${\bf Y}$ holds: $$ \sup_{x \in \mathbb{R}}|P(W_\theta \leq x) - \Phi(x)| \leq 2 \sum_{i=1}^n |\theta_i|^3 E| X_i|^3 + 8.4 E(V_\theta^2-1)^2, $$ where $Y_\theta=\theta \cdot {\bf Y}$ is the projection of ${\bf Y}$ in direction $\theta \in \mathbb{R}^n$ with $||\theta||=1$, $v_\theta=\sqrt{\mbox{Var}(Y_\theta)},X_i=|Y_i|/v_\theta$ and $V_\theta=\sum_{i=1}^n \theta_i^2 X_i^2$. As such coordinate symmetry arises in the study of projections of vectors chosen uniformly from the surface of convex bodies which have symmetries with respect to the coordinate planes, the main result is applied to a class of coordinate symmetric vectors which includes cone measure ${\cal C}_p^n$ on the $\ell_p^n$ sphere as a special case, resulting in a bound of order $\sum_{i=1}^n |\theta_i|^3$.

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Larry Goldstein. Qi-Man Shao. "Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors." Electron. Commun. Probab. 14 474 - 485, 2009. https://doi.org/10.1214/ECP.v14-1502

Information

Accepted: 30 October 2009; Published: 2009
First available in Project Euclid: 6 June 2016

zbMATH: 1189.60051
MathSciNet: MR2559097
Digital Object Identifier: 10.1214/ECP.v14-1502

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