Powers of large random unitary matrices and Toeplitz determinants

We study the limiting behavior of $\Tr U^{k(n)}$, where $U$ is a $n\times n$ 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 Szeg\"o limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence to this result, we find that for each fixed $m\in \N$, the random variables $\Tr U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,..., m$, converge to independent standard complex normals.