  We establish stability of random absolutely continuous invariant measures
(acims) for cocycles of random Lasota-Yorke maps under a variety of
perturbations. Our family of random maps need not be close to a fixed map;
thus, our results can handle very general driving mechanisms. We consider (i)
perturbations via convolutions, (ii) perturbations arising from finite-rank
transfer operator approximation schemes and (iii) static perturbations,
perturbing to a nearby cocycle of Lasota-Yorke maps. The former two results
provide a rigorous framework for the numerical approximation of random acims
using a Fourier-based approach and Ulam's method, respectively; we also
demonstrate the efficacy of these schemes.
