Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Characterization of a class of weak transport-entropy inequalities on the line

Nathael Gozlan, Cyril Roberto, Paul-Marie Samson, Yan Shu, and Prasad Tetali

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Abstract

We study an weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.

Résumé

Dans cet article, nous étudions une nouvelle famille de coûts de transport optimaux faibles en lien avec la notion d’ordre convexe pour les mesures de probabilité. Nous montrons, en dimension un, que le couplage optimal ne dépend pas de la fonction de coût choisie. Nous utilisons ensuite ce résultat pour établir une condition nécessaire et suffisante pour les inégalités de transport-entropie associées à ces coûts de transport faibles. En particulier, nous obtenons une forme transport équivalente de l’inégalité de Poincaré restreinte aux fonctions convexes sur la droite.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1667-1693.

Dates
Received: 13 January 2016
Revised: 6 June 2017
Accepted: 4 July 2017
First available in Project Euclid: 11 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296032

Digital Object Identifier
doi:10.1214/17-AIHP851

Mathematical Reviews number (MathSciNet)
MR3825894

Zentralblatt MATH identifier
06976088

Subjects
Primary: 60E15: Inequalities; stochastic orderings 32F32: Analytical consequences of geometric convexity (vanishing theorems, etc.) 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Transport inequalities Concentration of measure Majorization

Citation

Gozlan, Nathael; Roberto, Cyril; Samson, Paul-Marie; Shu, Yan; Tetali, Prasad. Characterization of a class of weak transport-entropy inequalities on the line. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1667--1693. doi:10.1214/17-AIHP851. https://projecteuclid.org/euclid.aihp/1531296032


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References

  • [1] R. Adamczak and M. Strzelecki. On the convex Poincaré inequality and weak transportation inequalities. Bernoulli. To appear, 2018. Available at arXiv:1703.01765v2.
  • [2] R. Adamczak and M. Strzelecki. Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions. Studia Math. 230 (1) (2015) 59–93.
  • [3] R. Adamczak. Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses. Bull. Pol. Acad. Sci. Math. 53 (2) (2005) 221–238.
  • [4] L. Ambrosio, N. Gigli and G. Savaré. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195 (2) (2014) 289–391.
  • [5] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer. Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses [Panoramas and Syntheses], 10. Société Mathématique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux.
  • [6] A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler. Oriented matroids, 2nd edition. Encyclopedia of Mathematics and its Applications 46. Cambridge University Press, Cambridge, 1999.
  • [7] S. G. Bobkov, I. Gentil and M. Ledoux. Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80 (7) (2001) 669–696.
  • [8] S. G. Bobkov and F. Götze. Discrete isoperimetric and Poincaré-type inequalities. Probab. Theory Related Fields 114 (2) (1999) 245–277.
  • [9] S. G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1) (1999) 1–28.
  • [10] S. G. Bobkov and C. Houdré. Weak dimension-free concentration of measure. Bernoulli 6 (4) (2000) 621–632.
  • [11] S. G. Bobkov and M. Ledoux. Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 (3) (1997) 383–400.
  • [12] V. I. Bogachev. Measure theory. Vol. I. Springer-Verlag, Berlin, 2007.
  • [13] S. Boucheron, G. Lugosi and P. Massart. Concentration inequalities. A nonasymptotic theory of independence. Oxford University Press, Oxford, 2013. With a foreword by Michel Ledoux.
  • [14] S. Cambanis, G. Simons and W. Stout. Inequalities for $Ek(X,Y)$ when the marginals are fixed. Z. Wahrsch. Verw. Gebiete 36 (4) (1976) 285–294.
  • [15] G. Dall’Aglio. Sugli estremi dei momenti delle funzioni di ripartizione doppia. Ann. Sc. Norm. Super. Pisa (3) 10 (1956) 35–74.
  • [16] H. Djellout, A. Guillin and L. Wu. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (2004) 2702–2732.
  • [17] R. M. Dudley. Real analysis and probability. Cambridge Studies in Advanced Mathematics 74. Cambridge University Press, Cambridge, 2002.
  • [18] N. Feldheim, A. Marsiglietti, P. Nayar and J. Wang. A note on the convex infimum convolution inequality. Bernoulli. To appear, 2018. Available at arXiv:1505.00240.
  • [19] M. Fréchet. Sur les tableaux dont les marges et des bornes sont données. Rev. Inst. Int. Stat. 28 (1960) 10–32.
  • [20] N. Gozlan. A characterization of dimension-free concentration in terms of transportation inequalities. Ann. Probab. 37 (6) (2009) 2480–2498.
  • [21] N. Gozlan. Transport-entropy inequalities on the line. Electron. J. Probab. 17 (49) (2012) 18.
  • [22] N. Gozlan and C. Léonard. Transport inequalities. A survey. Markov Process. Related Fields 16 (4) (2010) 635–736.
  • [23] N. Gozlan, C. Roberto and P.-M. Samson. A new characterization of Talagrand’s transport-entropy inequalities and applications. Ann. Probab. 39 (3) (2011) 857–880.
  • [24] N. Gozlan, C. Roberto and P.-M. Samson. Hamilton Jacobi equations on metric spaces and transport entropy inequalities. Rev. Mat. Iberoam. 30 (1) (2014) 133–163.
  • [25] N. Gozlan, C. Roberto, P.-M. Samson and P. Tetali. Kantorovich duality for general transport costs and applications. J. Funct. Anal. 272 (11) (2017) 3327–3405.
  • [26] G. H. Hardy, J. E. Littlewood and G. Pólya. Some simple inequalities satisfied by convex function. Messenger Math. 58 (1929) 145–152.
  • [27] J. B. Hiriart-Urruty and C. Lemaréchal. Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, 2001.
  • [28] F. Hirsch, C. Profeta, B. Roynette and M. Yor. Peacocks and associated martingales, with explicit constructions. Bocconi & Springer Series 3. Springer, Milan; Bocconi University Press, Milan, 2011.
  • [29] W. Hoeffding. Maßstabinvariante korrelationstheorie. Schr. Math. Inst. Inst. Angew. Math. Univ. Berlin 5 (1940) 181–233.
  • [30] W. Johnson and G. Schechtman. Remarks on Talagrand’s deviation inequality for Rademacher functions. Functional analysis. In Lecture Notes in Math. 72–77. Austin, TX, 1987/1989. Longhorn Notes 1470. Springer, Berlin, 1991.
  • [31] M. Ledoux. The concentration of measure phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI, 2001.
  • [32] J. Lott and C. Villani. Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. (9) 88 (3) (2007) 219–229.
  • [33] A. W. Marshall, I. Olkin and B. C. Arnold. Inequalities: theory of majorization and its applications, 2nd edition. Springer Series in Statistics. Springer, New York, 2011.
  • [34] K. Marton. A simple proof of the blowing-up lemma. IEEE Trans. Inform. Theory 32 (3) (1986) 445–446.
  • [35] K. Marton. Bounding $\bar{d}$-distance by informational divergence: A method to prove measure concentration. Ann. Probab. 24 (2) (1996) 857–866.
  • [36] K. Marton. A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (3) (1996) 556–571.
  • [37] K. Marton. An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces. J. Funct. Anal. 264 (1) (2013) 34–61.
  • [38] B. Maurey. Some deviation inequalities. Geom. Funct. Anal. 1 (2) (1991) 188–197.
  • [39] R. Rado. An inequality. J. Lond. Math. Soc. (2) 27 (1952) 1–6.
  • [40] P.-M. Samson. Concentration inequalities for convex functions on product spaces. In Stochastic inequalities and applications 33–52. Progr. Probab. 56. Birkhäuser, Basel, 2003.
  • [41] Y. Shu. From Hopf–Lax formula to optimal weak transfer plan. ArXiv preprint, 2016. Available at arXiv:1609.03405v1.
  • [42] Y. Shu and M. Strzelecki. A characterization of a class of convex log-Sobolev inequalities on the real line. Ann. Inst. Henri Poincaré B, Probab. Stat. To appear, 2018. Available at arXiv:1702.04698v1.
  • [43] S. Volker. The existence of probability measures with given marginals. Ann. Math. Stat. 36 (1965) 423–439.
  • [44] M. Strzelecka, M. Strzelecki and T. Tkocz. On the convex infimum convolution inequality with optimal cost function. ArXiv preprint, 2017. Available at arXiv:1702.07321v1.
  • [45] M. Talagrand. An isoperimetric theorem on the cube and the Kintchine–Kahane inequalities. Proc. Amer. Math. Soc. 104 (3) (1988) 905–909.
  • [46] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. Inst. Hautes Études Sci. 81 (1995) 73–205.
  • [47] M. Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (3) (1996) 587–600.
  • [48] C. Villani. Optimal transport: Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin, 2009.
  • [49] N.-Y. Wang. Concentration inequalities for Gibbs sampling under $d_{l_{2}}$-metric. Electron. Commun. Probab. 19 (63) (2014) 11.
  • [50] L. Wu. Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition. Ann. Probab. 34 (5) (2006) 1960–1989.