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.
The novelty here is to prove the Gaussian concentration bound for potentials with a variation decaying subexponentially. Let us know briefly explain what this means geometrically'. It is well-known that a uniformly expanding map @math of the unit interval with a finite Markov partition which is piecewise @math , for some @math , can be coded by a subshift of finite type @math over a finite alphabet. Then, @math induces a potential @math on @math which is Lipschitz. The pullback of @math is then the unique absolutely continuous invariant probability measure for @math . In @cite_8 , the authors showed that, given @math which is not Lipschitz, one can construct a uniformly expanding map of the unit interval with a finite Markov partition which is piecewise @math , but not piecewise @math for any @math , and such that the pullback of @math is the Lebesgue measure.
@cite_8
1994286529
We show how to construct a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. In particular we characterize the maps corresponding to stochastic chains with memory of variable length. The problem treated here is the converse of the classical construction of the Gibbs formalism for Markov expanding maps of the interval.