On the almost sure convergence of some ergodic means

We study the asymptotic behavior of ergodic averages of expectation of some randomly selected group of unitary operators, showing the mean convergence when the sequence of selectors is a ℤ d -valued random walk. We make use of the spectral decomposition of the unitary group to investigate the more difficult problem of almost sure convergence, and provide sufficient spectral conditions which carry out the almost everywhere convergence of these means when the sequence of selectors is a ℤ d -valued random walk satisfying some integrability conditions. We also show that this condition is optimal for d = 1, and deduce a speed of convergence for these averages using a Rademacher-Menchoff theorem on orthogonal series.