Rates of convergence in CLT and ASIP for sequences of expanding maps

We prove Berry-Esseen theorems and the almost sure invariance principle with rates for partial sums of the form $S_n={\sum }_{j=0}^{n-1}{f}_j\circ T_{j-1}\circ \cdots \circ T_1\circ T_0$ where $f_j$ are functions with uniformly bounded ``variation" and $T_j$ is a sequence of expanding maps. Using symbolic representations similar result follow for maps $T_j$ in a small $C^1$ neighborhood of an Axiom A map and Hölder continuous functions $f_j$. All of our results are already new for a single map $T_j=T$ and a sequence of different functions $(f_j)$.