Gaussian Concentration bound for potentials satisfying Walters condition with subexponential continuity rates - Archive ouverte HAL
Article Dans Une Revue Nonlinearity Année : 2019

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

J Moles
  • Fonction : Auteur
E Ugalde
  • Fonction : Auteur
  • PersonId : 1054426

Résumé

We consider the full shift $T:\Omega\to\Omega$ where $\Omega=A^{\mathbb{N}}$, $A$ being a finite alphabet. For a class of potentials which contains in particular potentials $\phi$ with variation decreasing like $O(n^{-\alpha})$ for some $\alpha>2$, we prove that their corresponding equilibrium state $\mu_\phi$ 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 $K(x,\ldots,T^{n-1}x)-\int K(y,\ldots,T^{n-1}y)\, \mathrm{d}\mu_\phi(y)$ 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 $\mu_\phi$. We also derive an almost-sure central limit theorem.
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Dates et versions

hal-02292419 , version 1 (19-09-2019)

Identifiants

  • HAL Id : hal-02292419 , version 1

Citer

J.-R Chazottes, J Moles, E Ugalde. Gaussian Concentration bound for potentials satisfying Walters condition with subexponential continuity rates. Nonlinearity, 2019. ⟨hal-02292419⟩

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