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|⁴ ≤ δ⁴. We observe that there exist many choices of coefficients θ₁,…,θₙ ∈ R with ∑ⱼ θⱼ² = 1 for which $_{α, β ∈ { R} {α < β}} | { P} ( α ≤ ∑ⱼ₌₁^n θ_j X_j ≤ β ) -({√2 π})-1/2 ∫αβ e-t^2/2\,dt | ≤ {C δ^4}/{n},$ 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 $O( 1/ √n)$ for the unit vector $θ = (1,…,1) / √n$. An explicit, universal example for coefficients $θ = (θ_1,…,θ_n)$ 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 $δ^4 = n⁻¹ ∑ⱼ₌₁^n { E}\, |X_j|^4$ 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.