VARIATIONS ON THE BERRY-ESSEEN THEOREM

Suppose that X-1,. . ., X-n are independent, identically distributed random variables of mean zero and variance one. Assume that E vertical bar X-1 vertical bar(4) <= delta(4). We observe that there exist many choices of coefficients theta(1),. . ., theta(n). is an element of R with root(j)theta(2)(j) = 1 for which sup(alpha<beta)alpha,beta is an element of R vertical bar P (alpha <= Sigma(n)(j=1) theta X-j(j) <= beta) - (root 2 pi(-1/2) f(alpha)(beta) e-(t2/2) dt vertical bar <= C delta(4)/n, where C > 0 is a universal constant. This inequality should be compared with the classical Berry- Esseen theorem, according to which the left- hand side may decay with n at the slower rate of O(1/root n) for the unit vector 0 - (1,. . ., 1)/root n. An explicit, universal example for coefficients theta = (theta(1),. . ., theta(n)) for which this inequality holds is theta = ( 1,root 2, -1, -root 2, -1, root 2, -1, -root 2,. . .) (3n/2)(-1/2), when n is divisible by four. Parts of the argument are applicable also in the more general case, in which X-1,. . ., X-n are independent random variables of mean zero and variance one yet are not necessarily identically distributed. In this general setting, the bound above holds with delta(4) = n(-1) Sigma(n)(j)=1 E vertical bar X-j vertical bar(4) for most selections of a unit vector theta = (theta(1),. . ., theta(n)) is an element of R-n. Here "most" refers to the uniform probability measure on the unit sphere.