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.
Let us also mention the paper @cite_2 in which the authors prove a Gaussian concentration bound for a @math which is attractive and of summable variation (whereas we need a bit more than summable). Their proof is based on coupling. However, they consider functions @math on @math , not on @math as in this paper. For such functions, the analogue of @math is @math . It is clear that a Gaussian concentration bound for functions @math implies a Gaussian concentration bound for functions @math , but the converse is not true.
@cite_2
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We prove that uniqueness of the stationary chain, or equivalently, of the @math -measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an i.i.d. process with countable alphabet, (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.