BERRY-ESSEEN BOUNDS FOR PROJECTIONS OF COORDINATE SYMMETRIC RANDOM VECTORS

For a coordinate symmetric random vector (Y-1, ... , Y-n) = Y is an element of R-n, that is, one satisfying (Y-1, ... ,Y-n) =(d) (e(1)Y(1), ... , e(n)Y(n)) for all (e(1), ... , e(n)) is an element of {-1,1}(n), for which P(Y-i = 0) = 0 for all i = 1,2, ... , n, the following Berry Esseen bound to the cumulative standard normal Phi for the standardized projection W-theta = Y-theta/nu(theta) of Y holds: sup(x is an element of R) vertical bar P(W-theta <= x) - Phi(x) vertical bar <= 2 Sigma(n)(i=1)vertical bar theta(i)vertical bar E-3 vertical bar X-i vertical bar(3) + 8.4E(V-theta(2) - 1)(2), where Y-theta = theta . Y is the projection of Y in direction theta is an element of R-n with parallel to theta parallel to = 1, nu(theta) = root Var(Y-theta), X-i = vertical bar Y-i vertical bar/nu(theta) and V-theta = Sigma(n)(i=1) theta X-2(i)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 l(p)(n) on the l(p)(n) sphere as a special case, resulting in a bound of order Sigma(n)(i=1) vertical bar theta(i)vertical bar(3).