  We prove central limit theorems and related results for non-uniformly
expanding random dynamical systems and general random Gibbs measures under some
integrability conditions and some (relatively weak) mixing related assumptions
on the base map. Some of our conditions also apply to maps which only expand on
average, in an appropriate sense. This is the first set of conditions that
allows to treat general non-uniformly expanding cases. A crucial step in our
proofs can be viewed as effective moderate versions of the random
Ruelle-Perron-Frobenius (RPF) theorem for the random transfer operator cocycle
corresponding to the random Gibbs measures (namely, effective moderate
estimates on the rate of "convergence" to the projection on the top Lyapunov
space in the corresponding multiplicative ergodic theorem). As a byproduct of
our methods we are also able to prove statistical properties of the
corresponding skew products and some estimates on the tails of random mixing
times. Compared with recent progress [40] on the subject, we are able to treat
random Gibbs measures corresponding to potentials which do not necessarily have
small local oscillation along inverse branches. Moreover, we are able to
consider more general classes of random expanding maps with unbounded
mixing/covering times and (only) local pairing property. Thus, our results
apply to a wide family of C^2-random expanding maps on compact C^2 manifolds,
together with the random geometric potential which corresponds to the unique
equivariant family of measures which are absolutely continuous with respect to
the volume measure. Other applications include high dimensional piecewise
expanding maps and random subshifts of finite type, which have applications to
random Markov interval maps and finite state Markov chains in random dynamical
environment.
