Powers of large random unitary matrices and Toeplitz determinants

We study the limiting behavior of $\operatorname{Tr}U^{k(n)}$, where $U$ is an $n\times n$ random unitary matrix and $k(n)$ is a natural number that may vary with $n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence of this result, we find that for each fixed $m\in \mathbb{N}$, the random variables $\operatorname{Tr}U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,\ldots,m$, converge to independent standard complex normals.