The Empirical Distribution of Fourier Coefficients

Suppose X₁, X₂, ⋯ are independent, identically distributed complex-valued L² random variables with EX₁ = 0 and E(|X₁|²) = 1. Let Yₙₖ be the kth Fourier coefficient of X₁, ⋯, Xₙ: Yₙₖ = ∑ⁿⱼ₌₁ Xⱼ exp (2π(-1)1/2kjn). Let μₙ be the empirical distribution of -1/2Yₙₖ: k = 1, ⋯, n\. Then μₙ converges to the distribution of $U + iV$, where U and V are independent normal variables with mean 0 and variance 1/2. This theorem is derived from a similar result for the Fourier coefficients of random permutations of the coordinates of zⁿ, where zⁿ is a vector with n coordinates such that maxₖ|zⁿₖ| = o(n1/2), as n → ∞.