On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields
Many statistical applications require establishing central limit theorems for sums, integrals, or for quadratic forms of functions of a stationary process. A particularly important case is that of Appell polynomials, since the Appell expansion rank" determines typically the type of central limit theorem satisfied by these functionals. We review and extend here to multidimensional indices a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well.
We reveal a connection of the Brascamp-Lieb inequality with Skorokhod embedding. Error bounds for the inequality in terms of variance are also provided.
In this short note we give a refinement of Brascamp-Lieb in the style of Houdre-Kagan extension for Poincare inequality in one dimension. This is inspired by works of Helffer and Ledoux.
We continue our investigation of the intertwining relations for Markov semigroups and extend the results of [9] to multi-dimensional diffusions. In particular these formulae entail new functional inequalities of Brascamp-Lieb type for log-concave distributions and beyond. Our results are illustrated by some classical and less classical examples.
We propose a new Borell-Brascamp-Lieb inequality which leads to novel sharp Euclidean inequalities such as Gagliardo-Nirenberg-Sobolev inequalities in R^n and in the half-space R^n\_+. This gives a new bridge between the geometric pont of view of the Brunn-Minkowski inequality and the functional point of view of the Sobolev type inequalities. In this way we unify, simplify and results by S. Bobkov-M. Ledoux, M. del Pino-J. Dolbeault and B. Nazaret.
In this article, we derive a new covariance estimate. The estimate has a similar structure as the Brascamp-Lieb inequality and is optimal for ferromagnetic Gaussian measures. It can be naturally applied to deduce decay of correlations of lattice systems of continuous spins. We also discuss the relation of the new estimate with known estimates like a weighted estimate due to Helffer & Ledoux. The main ingredient of the proof of the new estimate is a directional Poincaré inequality which seems to be unknown.
The works of Bennett, Carbery, Christ, Tao and of Valdimarsson have clarified when equality holds in the Brascamp-Lieb inequality. Here we characterize the case of equality in the Geometric case of Barthe's reverse Brascamp-Lieb inequality.
We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.
In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behaviour for the laws of solutions to stochastic differential equations.
This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres and symmetric groups). Our aim is twofold. Firstly, we provide a general, unifying, framework based on Markov generators, in order to cover a variety of examples of interest going beyond previous investigations. Secondly, we put forward the combinatorial reasons for which unexpected exponents occur in these inequalities.
$L^2$-stability for the variance Brascamp-Lieb inequality
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Eötvös University, Institute of Mathematics（厄特沃什大学数学研究所）；MODAL’X, Université Paris Nanterre（MODAL’X，巴黎楠泰尔大学）
We explore the consequences of the so-called intertwinings between gradients and Markov diffusion operators on $R^d$ in terms of second-order Brascamp-Lieb inequalities for log-concave distributions and beyond, extending our inequalities established in a previous paper. As a result, we derive some convenient lower bounds on the $(d+1)^{th}$ positive eigenvalue depending on the spectral gap of the dual Markov diffusion operator given by the intertwining. To see the relevance of our approach, we apply our spectral results in the case of perturbed product measures, freeing us from Helffer's class…
We prove a general duality result showing that a Brascamp--Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This open a new approach to the proof of Brascamp--Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $\R^n$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of th…
In this paper, we derive sharp nonlinear dimension-free Brascamp--Lieb inequalities (including hypercontractivity inequalities) for distributions on Polish spaces, which strengthen the classic Brascamp--Lieb inequalities. Applications include the extension of Mrs. Gerber's lemma to the cases of Rényi divergences and distributions on Polish spaces, the strengthening of small-set expansion theorems, and the characterization of the exponent of the $q$-stability. Our proofs in this paper are based on information-theoretic and coupling techniques.
The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish …
We prove central limit theorems for additive functionals of stationary fields under integrability conditions on the higher-order spectral densities, which are derived using the Holder-Young-Brascamp-Lieb inequality.
In this paper, we prove that for the doubly symmetric binary distribution, the lower increasing envelope and the upper envelope of the minimum-relative-entropy region are respectively convex and concave. We also prove that another function induced the minimum-relative-entropy region is concave. These two envelopes and this function were previously used to characterize the optimal exponents in strong small-set expansion problems and strong Brascamp--Lieb inequalities. The results in this paper, combined with the strong small-set expansion theorem derived by Yu, Anantharam, and Chen (2021), and …
This work is devoted to the geometric analysis of metric-measure spaces satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb inequality. Completing the early investigations by Cordero-Erausquin, McCann and Schmuckenschlager, we show that these functional inequalities characterize lower bounds on the Ricci curvature on a Riemannian manifold, providing thus an alternate version of Ricci curvature lower bounds in measured length spaces to the recent developments by Lott, Villani and Sturm. We also investigate stability properties and geometric and functional inequalities, such as …
We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp--Lieb concentration inequality, and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the $\nablaφ$ type. The classical Brascamp--Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of c…
We provide variants and improvements of the Brascamp-Lieb variance inequality which take into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.
We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp-Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian Hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in literatures including the…
We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.
In 1998, Boué and Dupuis proved a variational representation for exponentials of bounded Wiener functionals. Since their proof involves arguments related to the weak convergence of probability measures, the boundedness of functionals seems inevitable. In this paper, we extend the representation to unbounded functionals under a mild assumption on their integrability. As an immediate application of the extension, we prove an analogue of Prékopa's theorem for Wiener functionals, which is then applied to formulate the Brascamp-Lieb inequality in the framework of Wiener spaces.
The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.
Brascamp--Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $κ$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.
We investigate the interactions of functional rearrangements with Prekopa-Leindler type inequalities. It is shown that that a general class of integral inequalities tighten on rearrangement to "isoperimetric" sets with respect to a relevant measure. Applications to the Borell-Brascamp-Lieb, Borell-Ehrhart, and the recent polar Prekopa-Leindler inequalities are demonstrated. It is also proven that an integrated form of the Gaussian log-Sobolev inequality decreases on half-space rearrangement.
We prove a new type of Poincaré inequality on abstract Wiener spaces for a family of probability measures which are absolutely continuous with respect to the reference Gaussian measure. This class of probability measures is characterized by the strong positivity (a notion introduced by Nualart and Zakai in [17]) of their Radon-Nikodym densities. Measures of this type do not belong in general to the class of log-concave measures, which are a wide class of measures satisfying the Poincaré inequality (Brascamp and Lieb [2]). Our approach is based on a point-wise identity relating Wick and ordinar…
We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained H…
