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The classical concentration inequalities deal with the deviations
of functions of independent and identically distributed (i.i.d.) random variables from their
expectation and these inequalities have numerous important applications in statistics and
machine learning theory. In this paper we go far beyond this classical framework by establish
two new Bernstein type concentration inequalities for -mixing sequence and uniformly
ergodic Markov chains. As the applications of the Bernstein's inequalities, we also obtain
the bounds on the rate of uniform deviations of empirical risk minimization (ERM) algorithms
based on -mixing observations.