Suppose $X_1, X_2, \cdots$ are independent, identically distributed complex-valued $L^2$ random variables with $EX_1 = 0$ and $E(|X_1|^2) = 1$. Let $Y_{nk}$ be the $k$th Fourier coefficient of $X_1, \cdots, X_n$: $Y_{nk} = \sum^n_{j=1} X_j \exp \big(\frac{2\pi(-1)^{1/2}kj}{n}\big).$ Let $\mu_n$ be the empirical distribution of $\{n^{-1/2}Y_{nk}: k = 1, \cdots, n\}$. Then $\mu_n$ converges to the distribution of $U + iV$, where $U$ and $V$ are independent normal variables with mean 0 and variance $\frac{1}{2}$. This theorem is derived from a similar result for the Fourier coefficients of random permutations of the coordinates of $z^n$, where $z^n$ is a vector with $n$ coordinates such that $\max_k|z^n_k| = o(n^{1/2})$, as $n \rightarrow \infty$.