Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods

We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad family of spectral regularization methods for reproducing kernel decision rules, when trained on a sample coming from a $τ-$mixing process.