  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 H\"older continuous functions $f_j$. All of
our results are already new for a single map $T_j=T$ and a sequence of
different functions $(f_j)$.
