4 Exactness of Rokhlin Endomorphisms and Weak Mixing of Poisson Boundaries

We give conditions for the exactness of Rokhlin skew products, apply these to random walks on locally compact, second countable topological groups and obtain that the Poisson boundary of a globally supported random walk on such a group is weakly mixing. We give (theorem 2.3) conditions for the exactness of the Rokhlin endomorphism T = These conditions are applied to random walk-endomorphisms. Meilijson (in [Me]) gave sufficient conditions for exactness for random walk-endomorphisms over G = Z. We clarify Meilijson's theorem, proving a converse (proposition 4.2), extend it to countable Abelian groups (theorem 4.1) and characterize the exactness of the Rokhlin endomorphism for an aperiodic random walk on a countable group (theorem 4.5). Tools employed include the ergodic theory of " associated actions " (see §1), and the boundary theory of random walks (see §4). As a spinoff we obtain that the right action on Poisson boundary (see §4) of a globally supported random walk is weakly mixing (proposition 4.4). §1 Associated actions For an endomorphism R of a measure space (Z, D, ν) set • I(R) := {A ∈ D : R −1 A = A} – the invariant σ-algebra, and • T (R) := ∞ n=0 R −n D – the tail σ-algebra.