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) 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.