Gaussian Concentration bound for potentials satisfying Walters condition with subexponential continuity rates

We consider the full shift $T:Ω\toΩ$ where $Ω=A^{\mathbb N}$, $A$ being a finite alphabet. For a class of potentials which contains in particular potentials $ϕ$ with variation decreasing like $O(n^{-α})$ for some $α>2$, we prove that their corresponding equilibrium state $μ_ϕ$ satisfies a Gaussian concentration bound. Namely, we prove that there exists a constant $C>0$ such that, for all $n$ and for all separately Lipschitz functions $K(x_0,\ldots,x_{n-1})$, the exponential moment of $K(x,\ldots,T^{n-1}x)-\int K(y,\ldots,T^{n-1}y)\, \mathrm{d}μ_ϕ(y)$ is bounded by $\exp\big(C\sum_{i=0}^{n-1}\mathrm{Lip}_i(K)^2\big)$. The crucial point is that $C$ is independent of $n$ and $K$. 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 $μ_ϕ$. We also derive an almost-sure central limit theorem.

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