  In this paper we obtain an almost sure invariance principle for convergent
sequences of either Anosov diffeomorphisms or expanding maps on compact
Riemannian manifolds and prove an ergodic stability result for such sequences.
The sequences of maps need not correspond to typical points of a random
dynamical system. The methods in the proof rely on the stability of
compositions of hyperbolic dynamical systems. We introduce the notion of
sequential conjugacies and prove that these vary in a Lipschitz way with
respect to the generating sequences of dynamical systems. As a consequence, we
prove stability results for time-dependent expanding maps that complement
results in [Franks74] on time-dependent Anosov diffeomorphisms.
