Convergence rates in the law of large numbers for Banach-valued dependent variables

We extend Marcinkievicz–Zygmund strong laws of large numbers for martingales to weakly dependent random variables with values in smooth Banach spaces. The conditions are expressed in terms of conditional expectations. In the case of Hilbert spaces, we show that our conditions are weaker than optimal ones for strongly mixing sequences (which were previously known for real-valued variables only). As a consequence, we give rates of convergence for Cramér–von Mises statistics and for the empirical estimator of the covariance operator of a Hilbert-valued autoregressive process.