Recurrence of cocycles and stationary random walks

We survey distributional properties of R-valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in R) which determine recurrence or transience. Let (Xn, n ≥ 0) be an ergodic stationary Rd-valued stochastic process, and let (Yn = X0 + · · · + Xn−1, n ≥ 1) be the associated random walk. What can one say about recurrence of this random walk if one only knows the distributions of the random variables Yn, n ≥ 1? It turns out that methods from ergodic theory yield some general sufficient conditions for recurrence of such random walks without any assumptions on independence properties or moments of the process (Xn). Most of the results described in this note have been published elsewhere. Only the Theorems 12 and 14 on recurrence of symmetrized random walks are — to my knowledge — new. Let us start our discussion by formulating the recurrence problem in the language of ergodic theory. Let T be a measure preserving ergodic automorphism of a probability space (X, S, μ), d ≥ 1, and let f : X −→ R be a Borel map. The cocycle f : Z×X −→ Rd is given by