  We study mixing properties of generalized $T, T^{-1}$ transformation. We
discuss two mixing mechanisms. In the case the fiber dynamics is mixing, it is
sufficient that the driving cocycle is small with small probability. In the
case the fiber dynamics is only assumed to be ergodic, one needs to use the
shearing properties of the cocycle. Applications include the Central Limit
Theorem for sufficiently fast mixing systems and the estimates on deviations of
ergodic averages.
