On the convex Poincaré inequality and weak transportation inequalities

Adamczak, Radosław; Strzelecki, Michał

doi:10.3150/17-bej989

Cited by 10 publications

(6 citation statements)

References 29 publications

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“…In fact, weak transportation inequalities with such strengthened cost functions can hold only for compactly supported measures (see [38]). The interest in such strengthening lies in the fact that by taking into account the boundedness of random variables, it implies concentration inequalities stronger than the subgaussian bound given by (5.2) (see e.g., [4] for concentration results corresponding to various cost functions θ). As shown in the next proposition, such inequalities can be also easily inferred just at the level of convex concentration.…”

Section: Convex Concentrationmentioning

confidence: 99%

A note on concentration for polynomials in the Ising model

Adamczak¹,

Kotowski²,

Polaczyk³

et al. 2019

Electron. J. Probab.

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We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson-Wright type inequality.We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.2010 Mathematics Subject Classification. 60E15, 82B99.

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Section: Convex Concentrationmentioning

confidence: 99%

A note on concentration for polynomials in the Ising model

Adamczak¹,

Kotowski²,

Polaczyk³

et al. 2019

Electron. J. Probab.

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“…It is an interesting question what moment estimates can be obtained under an additional convexity assumption. We remark that for the usual notion of convexity on R n , certain self-normalized moment estimates have been derived for all measures satisfying the convex concentration property [5].…”

Section: Proposition 418 Assume Thatmentioning

confidence: 99%

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Adamczak,

Polaczyk,

Strzelecki

2020

Preprint

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We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as p → 1 + are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities.We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space. Contents1. Introduction 1.1. Motivation and informal presentation 1.2. General setting 1.3. Examples 1.4. Functional inequalities 2. From modified log-Sobolev to Beckner's inequalities 2.1. Main result 2.2. Auxiliary lemmas 2.3. Proof of Theorem 2.1 3. Moment estimates derived from Beckner's inequalities 4. Applications 4.1. The continuous setting 4.2. Product spaces 4.3. Glauber dynamics 4.4. The symmetric group 4.5. Stochastic covering property 4.6. Zero-range processes 4.7. The Poisson space 5. Higher order concentration 5.1. Abstract inequality 5.2. Discussion on the choice of gradients 5.3. Applications to tetrahedral polynomials Appendix A.

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“…The inequality is known to specialists and the method of proof goes at least back to [BG99] and [Sam03]. The inequality is weaker than (5.1) but holds for a larger class of measures: for all measures which satisfy a quadratic cost inequality á la Talagrand [Tal96] (see also [AS17] for a recent development on the related subject). First we recall the necessary definitions.…”

Section: 2mentioning

confidence: 99%

Variance estimates and almost Euclidean structure

Paouris

Valettas

2019

Advances in Geometry

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We introduce and initiate the study of new parameters associated with any norm and any log-concave measure on R n , which provide sharp distributional inequalities. In the Gaussian context this investigation sheds light to the importance of the statistical measures of dispersion of the norm in connection with the local structure of the ambient space. As a byproduct of our study, we provide a short proof of Dvoretzky's theorem which not only supports the aforementioned significance but also complements the classical probabilistic formulation.

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On the convex Poincaré inequality and weak transportation inequalities

Cited by 10 publications

References 29 publications

A note on concentration for polynomials in the Ising model

A note on concentration for polynomials in the Ising model

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Variance estimates and almost Euclidean structure

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