ELECTRONIC COMMUNICATIONS in PROBABILITY BERRY-ESSEEN BOUNDS FOR PROJECTIONS OF COORDINATE SYMMETRIC RANDOM VECTORS

convex bodies 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 fol-lowing Berry Esseen bound to the cumulative standard normal Φ for the standardized projection Wθ = Yθ/vθ of Y holds: sup x∈R |P(Wθ ≤ x)−Φ(x) | ≤ 2 n∑ i=1 |θi |3E|X i |3 + 8.4E(V 2θ − 1)2, where Yθ = θ · Y is the projection of Y in direction θ ∈ Rn with ||θ | | = 1, vθ = p Var(Yθ),X i