  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 $\tau-$mixing
process.
