Powers of large random unitary matrices and Toeplitz determinants

We study the limiting behavior of TrUk(n), where U is a n × n random unitary matrix and k(n) is 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 to this result, we find that for each fixed m ∈ N, the random variables TrUkj(n)/ √ min(kj(n), n), j = 1, . . . ,m, converge to independent standard complex normals.