Central limit theorem under the Dedecker-Rio condition in some Banach spaces

We extend the central limit theorem under the Dedecker-Rio condition to adapted stationary and ergodic sequences of random variables taking values in a class of smooth Banach spaces. This result applies to the case of random variables taking values in $L^p(\mu )$, with $2\le p<\mathrm{\infty }$ and $\mu$ a $\sigma$-finite real measure. As an application we give a sufficient condition for empirical processes indexed by Sobolev balls to satisfy the central limit theorem, and discuss about the optimality of these conditions.