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 Electronic Communications in Probability > Vol. 14 (2009) > Paper 46 open journal systems 


Berry-Esseen Bounds for Projections of Coordinate Symmetric Random Vectors

Larry Goldstein, University of Southern California
Qi-Man Shao, Hong Kong University of Science and Technology


Abstract
For a coordinate symmetric random vector (Y1,...,Yn) = YRn, that is, one satisfying (Y1,...,Yn) =d (e1Y1,...,enYn) for all (e1,..,en) ∈ {-1,1}n, for which P(Yi=0)=0 for all  i=1,2,...,n, the following Berry Esseen bound to the cumulative standard normal Φ for the standardized projection Wθ=Yθ/vθ of Y holds:

supx ∈ R|P(Wθ ≤ x) - Φ(x)| ≤   2 ∑i=1,...,ni|3 E|Xi|3 + 8.4 E(Vθ2-1)2,

where Yθ =θ • Y is the projection of Y in direction θ ∈ Rn with ||θ||=1, vθ=√Var(Yθ), Xi=|Yi|/vθ and Vθ=∑i=1,...,n θi2 Xi2. 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 Cpn on the lpn sphere as a special case, resulting in a bound of order ∑i=1,...,ni|3.


Full text: PDF

Pages: 474-485

Published on: October 30, 2009


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Electronic Communications in Probability. ISSN: 1083-589X