## Electronic Journal of Probability

### Non asymptotic variance bounds and deviation inequalities by optimal transport

Kevin Tanguy

#### Abstract

The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\mathbb{R} ^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 13, 18 pp.

Dates
Accepted: 5 January 2019
First available in Project Euclid: 20 February 2019

https://projecteuclid.org/euclid.ejp/1550653272

Digital Object Identifier
doi:10.1214/19-EJP265

Mathematical Reviews number (MathSciNet)
MR3916333

Zentralblatt MATH identifier
07055651

#### Citation

Tanguy, Kevin. Non asymptotic variance bounds and deviation inequalities by optimal transport. Electron. J. Probab. 24 (2019), paper no. 13, 18 pp. doi:10.1214/19-EJP265. https://projecteuclid.org/euclid.ejp/1550653272

#### References

• [1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G.Scheffer. Sur les inégalités de Sobolev logarithmiques. Société mathématiques de France, 2000.
• [2] D. Bakry, I. Gentil, and M. Ledoux. Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften, 348, 2014.
• [3] F. Barthe and C. Roberto. Modified logarithmic Sobolev inequalities on $\mathbb{R}$. Potential Analysis, 2008.
• [4] S. Bobkov. Isoperimetric Inequalities for Distributions of Exponential Type. The Annals of Probability, Vol. 22, No 2, 978-994, 1994.
• [5] S. Bobkov and C. Houdré. A converse Gaussian Poincaré-type inequality for convex functions. Statistics and Probability Letters, 44(3):281–290, 1999.
• [6] S. Bobkov and M. Ledoux. Weighted Poincare-Type Inequalities For Cauchy And Other Convex Measures. The Annals of Probability, 37(2):403–427, 2009.
• [7] S. Boucheron and M. Thomas. Concentration inequalities for order statistics. Electronic Communications in Probability, 2012.
• [8] T. Boucheron, G. Lugosi, and P. Massart. Concentration inequalities : a nonasymptotic theory of independance. Oxford University Press, 2013.
• [9] D. Chafaï and S. Péché. A note on the second order universality at the edge of Coulomb gases on the plane. J. Stat. Phys., 2:368–383, 2014.
• [10] S. Chatterjee. Superconcentration and related topics. Springer, 2014.
• [11] M. Damron, J. Hanson, and P. Sosoe. Subdiffusive concentration in first-passage percolation. Electronic Journal of Probability, 19(109), 2014.
• [12] M. Damron, J. Hanson, and P. Sosoe. Sublinear variance in first-passage percolation for general distributions. Probab. Theory Related Fields, 163(1-2):223–258, 2015.
• [13] L. De Haan and A. Ferreira. Extreme Value Theory. Springer Series in Operations Research and Financial Engineering, 2006.
• [14] N. Gozlan. Poincaré inequalities and dimension free concentration of measure. Ann. Inst. Henri Poincaré Prob. Stat. 46 (2010), no. 3, 2010.
• [15] M. R. Leadbetter, G. Lindgren, and H. Rootzén. Extremes and related properties of random sequences and processes. Springer Series in Statistics., 1983.
• [16] M. Ledoux. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89, 2001.
• [17] G. Paouris, P. Valettas, and J. Zinn. Random version of Dvoretzky’s Theorem in $l_p^n$. 2015.
• [18] B. Rider. Order statistics and Ginibre’s ensembles. J. Stat. Phys., 114(3-4):1139–1148, 2004.
• [19] G. Schechtman. The random version of Dvoretsky’s theorem in $l^n_\infty$. GAFA Seminar 2004-2005, 1910:265–270, 2007.
• [20] M. Talagrand. On Russo’s approximate zero-one law. Ann. Prob. 22, 1576-1587,, 1994.
• [21] M. Talagrand. An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc. 104, 905-909, 1998.
• [22] K. Tanguy. Some superconcentration inequalities for extrema of stationary gaussian processes. Statistics and Probability Letters, 2015.
• [23] K. Tanguy. Quelques inégalités de superconcentration : théorie et applications (in french). PhD thesis, Institute of Mathematics of Toulouse, 2017.
• [24] P. Valettas. On the tightness of Gaussian concentration for convex functions. Journal d’Analyse Mathématique, To appear.
• [25] C. Villani. Topics in Optimal Transportation. AMS, 2003.