F eb 2 01 0 Variations on the Berry - Esseen theorem

Suppose that X1, . . . ,Xn are independent, identically-distributed random variables of mean zero and variance one. Assume that E|X1| ≤ δ4. We observe that there exist many choices of coefficients θ1, . . . , θn ∈ R with ∑ j θ 2 j = 1 for which sup α,β∈R α<β ∣∣∣∣∣ P  α ≤ n ∑ j=1 θjXj ≤ β   − 1 √ 2π ∫ β α e−t 2/2dt ∣∣∣∣∣ ≤ Cδ 4 n , (1) where C > 0 is a universal constant. Inequality (1) should be compared with the classical Berry-Esseen theorem, according to which the left-hand side of (1) may decay with n at the slower rate of O(1/ √ n), for the unit vector θ = (1, . . . , 1)/ √ n. An explicit, universal example for coefficients θ = (θ1, . . . , θn) for which (1) holds is θ = (1, √ 2,−1,− √ 2, 1, √ 2,−1,− √ 2, · · · ) /√ 3n/2 when n is divisible by four. Parts of the argument are applicable also in the more general case, in whichX1, . . . ,Xn are independent random variables of mean zero and variance one, yet they are not necessarily identically distributed. In this general setting, the bound (1) holds with δ4 = n−1 ∑n j=1 E|Xj |4 for most selections of a unit vector θ = (θ1, . . . , θn) ∈ Rn. Here “most” refers to the uniform probability measure on the unit sphere.