A note on the random mean ergodic theorem

Abstract

Let {Sn} be any non-trivial random walk on the integers. Then, almost surely, for all p≧1, for any probability space X, for any mixing automorphism T on X, for all feLp(X), $$\frac{1}{n}\sum\limits_{i = 1}^n {f \circ T^{S_i } \to E(f) in L_p (X).}$$