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) = Y∈ Rn, 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,...,n |θi|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,...,n |θi|3.
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