Approximating class approach for empirical processes of dependent sequences indexed by functions

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class ${\mathcal{F}}$ which fits this setting. Our results apply to the empirical process of data $(X_i)_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.