  We discuss recurrence and ergodicity properties of random walks and
associated skew products for large classes of locally compact groups and
homogeneous spaces. In particular we show that a closed subgroup of a product
of finitely many linear groups over local fields supports a recurrent random
walk if and only if it has at most quadratic growth. We give also a detailed
analysis of ergodicity properties for special classes of random walks on
homogeneous spaces. The structure of closed subgroups of linear groups over
local fields and the properties of group actions with respect to stationary
measures play an important role in the proofs.
