We consider the full shift @math where @math , @math being a finite alphabet. For a class of potentials which contains in particular potentials @math with variation decreasing like @math for some @math , we prove that their corresponding equilibrium state @math satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant @math such that, for all @math and for all separately Lipschitz functions @math , the exponential moment of @math is bounded by @math . The crucial point is that @math is independent of @math and @math . We then derive various consequences of this inequality. For instance, we obtain bounds on the fluctuations of the empirical frequency of blocks, the speed of convergence of the empirical measure, and speed of Markov approximation of @math . We also derive an almost-sure central limit theorem.
For @math is Lipschitz, Theorem was proved in @cite_7 . The main goal of @cite_7 was then to deal with nonuniformly hyperbolic systems modeled by a Young tower. For maps of the unit interval, one can have for instance maps with an indifferent fixed point. When the tower has exponential tails, the authors of @cite_7 proved a Gaussian concentration bound. When the tower has polynomial tails, they proved moment concentration bounds.
@cite_7
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For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.