Variance of sum of random roots of unity

Question

Let $X_{m,n}$ be the sum of $n$ uniformly random, independent samples from the set of all $m$th roots of unity. Obviously, the expectation of $X_{m,n}$ is $0$. How do I go about reasoning about the variance as a function of $m,n$? Crucially, does the variance grow more slowly if $m$ is large, or does $X_{m,n}$ behave the same as $X_{2,n}$ (random coin tosses)?

Answer 1

Let $C_m=\sum_{k=0}^{m-1} \cos^2(2\pi k/m)\big/ m$. The your desired covariance is $$n \begin{pmatrix}C_m&0\\0&1-C_m\end{pmatrix}.$$ For big $m$ the quantity $C_m\approx 1/2$.