Spectral methods for limit theorems for random expanding transformations

We extend the spectral method for proving limit theorems to random non-uniformly expanding dynamical systems. This yields the CLT and moderate deviations principles (MDP). We show that as the amount of non-uniformity decreases the CLT rates and the speed in the MDP become closer to the optimal ones. For smooth systems the rates are effective. Compared to recent progress on the subject [34] we are able to consider much more general maps, Gibbs measures and observables. However, our main results are new even in the setup of [34].