  We establish a Bernstein-type inequality for a class of stochastic processes
that include the classical geometrically $\phi$-mixing processes, Rio's
generalization of these processes, as well as many time-discrete dynamical
systems. Modulo a logarithmic factor and some constants, our Bernstein-type
inequality coincides with the classical Bernstein inequality for i.i.d.~data.
We further use this new Bernstein-type inequality to derive an oracle
inequality for generic regularized empirical risk minimization algorithms and
data generated by such processes. Applying this oracle inequality to support
vector machines using the Gaussian kernels for both least squares and quantile
regression, it turns out that the resulting learning rates match, up to some
arbitrarily small extra term in the exponent, the optimal rates for
i.i.d.~processes.
