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

## TL;DR

This paper proves a Gaussian concentration bound for equilibrium states of certain potentials with Walters condition, leading to new results on fluctuations, convergence rates, and an almost-sure CLT in symbolic dynamics.

## Contribution

It establishes a Gaussian concentration inequality for a class of potentials with subexponential variation decay, independent of the Lipschitz functions involved.

## Key findings

- Bound on fluctuations of empirical frequencies
- Speed of convergence of empirical measures
- Almost-sure central limit theorem

## Abstract

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.