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

A Bernstein-type Inequality for Some Mixing Processes and Dynamical Systems with an Application to Learning