Sharp ergodic theorems for group actions and strong ergodicity

Let μ be a probability measure on a locally compact group G, and suppose G acts measurably on a probability measure space (X,m), preserving the measure m. We study ergodic theoretic properties of the action along μ-i.i.d. random walks on G. It is shown that under a (necessary) spectral assumption on the μ-averaging operator on L2(X,m), almost surely the mean and the pointwise (Kakutani’s) random ergodic theorems have roughly n−1/2 rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonalG-action on (X × X,m × m), an almost surely exponential rate of mixing along random walks is obtained. The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of m-integration as a G-invariant mean on L∞(X,m). These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable G. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups’ homogeneous spaces, for which our results can be applied.