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Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors
2.	Creator	Author's name, affiliation, country	Larry Goldstein; University of Southern California
2.	Creator	Author's name, affiliation, country	Qi-Man Shao; Hong Kong University of Science and Technology
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	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 $\sum_{i=1}^n |\theta_i|^3$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2009-10-30
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1502
10.	Identifier	Digital Object Identifier	10.1214/ECP.v14-1502
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 14
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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