  We prove a Chung-Fuchs type theorem for skew product dynamical systems such
that for a measurable function on such a system, if its Birkhoff average
converges to zero almost surely, and on typical fibres its Birkhoff sums have a
non-trivial independent structure, then its associated generalised random walk
oscillates, that is the supremum of the random walk equals to $+\infty$ and the
infimum equals to $-\infty$.
