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We prove central limit theorems and related results for non-uniformlyexpanding random dynamical systems and general random Gibbs measures under someintegrability conditions and some (relatively weak) mixing related assumptionson the base map. Some of our conditions also apply to maps which only expand onaverage, in an appropriate sense. This is the first set of conditions thatallows to treat general non-uniformly expanding cases. A crucial step in ourproofs can be viewed as effective moderate versions of the randomRuelle-Perron-Frobenius (RPF) theorem for the random transfer operator cocyclecorresponding to the random Gibbs measures (namely, effective moderateestimates on the rate of "convergence" to the projection on the top Lyapunovspace in the corresponding multiplicative ergodic theorem). As a byproduct ofour methods we are also able to prove statistical properties of thecorresponding skew products and some estimates on the tails of random mixingtimes. Compared with recent progress [40] on the subject, we are able to treatrandom Gibbs measures corresponding to potentials which do not necessarily havesmall local oscillation along inverse branches. Moreover, we are able toconsider more general classes of random expanding maps with unboundedmixing/covering times and (only) local pairing property. Thus, our resultsapply 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 uniqueequivariant family of measures which are absolutely continuous with respect tothe volume measure. Other applications include high dimensional piecewiseexpanding maps and random subshifts of finite type, which have applications torandom Markov interval maps and finite state Markov chains in random dynamicalenvironment.

Effective (moderate) random RPF theorems and applications to limit theorems for non-uniformly expanding RDS