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
2950162031 2102318662 1493594095 2028204099
We investigate the clustering performances of the relaxed @math means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decay exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed @math means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed @math means SDP allows to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed @math means. We study the maximum of a Gaussian field on @math ( @math ) whose correlations decay logarithmically with the distance. Kahane Kah85 introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas DRSV12a DRSV12b extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in DRSV12a several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in DRSV12a : we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order @math @math of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order @math is finite, uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.