Mixing properties of induced random transformations

Let S(N) be a random walk on a countable abelian group G which acts on a probability space E by measure–preserving transformations (Tv)v∈G. For any Λ ⊂ E we consider the random return time τ at which TS(τ) ∈ Λ. We show that the corresponding induced skew product transformation is K–mixing whenever a natural subgroup of G acts ergodically on E.