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 $L^p\left(\mu \right)$, with $2⩽p<\infty$ 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.