Asymptotics of powers of random elements of compact Lie groups*

For a Haar -distributed element H of a compact Lie group L, Eric Rains proved in [10] that there is a natural number D = DL such that, for all d >= D, the eigenvalue distribution of Hd is fixed, and Rains described this fixed eigenvalue distribution explicitly. In the present paper we consider random elements U of a compact Lie group with general distribution. In particular, we introduce a mild absolute continuity condition under which the eigenvalue distribution of powers of U converges to that of HD. Then, rather than the eigenvalue distribution, we consider the limiting distribution of Ud itself.