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 \(-\infty \) . As an application we show that the fibre measures of the natural extension of ergodic measures on symbolic dynamical systems are degenerate with respect to the Mandelbrot cascade action in the critical situation.