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|X ₁| ⁴ ≦ δ ⁴. We observe that there exist many choices of coefficients θ ₁,..., θ _n ∈ R with Σ _j θ ² _j = 1 for which sup, 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/ √ n) for the unit vector θ = (1, ..., 1)/ √ n. An explicit, universal example for coefficients θ = (θ1, ..., θn) for which this inequality holds is θ = (1, √ 2,-1,- √ 2, 1, √ 2,-1,- √ 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 δ ⁴ = n ^-1Σ ⁿ _j=1 E|X _j | ⁴ for most selections of a unit vector θ = (θ ₁, ..., θ _n) ∈ R ⁿ. Here "most" refers to the uniform probability measure on the unit sphere.