Moment estimates implied by modified log-Sobolev inequalities

Adamczak, Radosław; Bednorz, Witold; Wolff, Paweł

ESAIM: Probability and Statistics, Tome 21 (2017), pp. 467-494.

We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on ${ℝ}^n$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-type estimates for deviations of smooth (not necessarily Lipschitz) functions and measures of enlargements of sets corresponding to non-Euclidean norms. We also prove a two-level concentration result for functions of bounded Hessian and measures satisfying the classical logarithmic Sobolev inequality.

R. Adamczak, Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses. Bull. Pol. Acad. Sci. Math. 53 (2005) 221–238. | DOI | MR | Zbl

R. Adamczak and R. Latała, Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 1103–1136. | DOI | Numdam | MR | Zbl

R. Adamczak and P. Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Related Fields 162 (2015) 531–586. | DOI | MR | Zbl

S. Aida and D. Stroock, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1 (1994) 75–86. | DOI | MR | Zbl

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les inégalités de Sobolev logarithmiques. Vol. 10 of Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris (2000). | MR | Zbl

M.A. Arcones and E. Giné, On decoupling, series expansions and tail behavior of chaos processes. J. Theor. Probab. 6 (1993) 101–122. | DOI | MR | Zbl

D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators. Springer, Cham (2014). | MR | Zbl

F. Barthe and A.V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18 (2008) 921–979. | DOI | MR | Zbl

F. Barthe and C. Roberto, Modified logarithmic Sobolev inequalities on $R$. Potential Anal. 29 (2008) 167–193. | DOI | MR | Zbl

W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 (1989) 397–400. | MR | Zbl

S. Bobkov, Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24 (1996) 35–48. | DOI | MR | Zbl

S. Bobkov, G. Chistyakov and F. Götze, Second order concentration on the sphere. Commun. Contemp. Math. 19 (2017) 1650058. | DOI | MR | Zbl

S. Bobkov and M. Ledoux, Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 (1997) 383–400. | DOI | MR | Zbl

S.G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1–28. | DOI | MR | Zbl

S.G. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000) 1028–1052. | DOI | MR | Zbl

S.G. Bobkov and B. Zegarlinski, Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 (2005) (829) . | MR | Zbl

C. Borell, On the Taylor series of a Wiener polynomial. Seminar Notes on multiple stochastic integration, polynomial chaos and their integration. Case Western Reserve Univ., Cleveland (1984).

P. Federbush, Partially alternate derivation of a result of Nelson. J. Math. Phys. 10 (1969) 5052. | DOI | Zbl

I. Gentil, From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality. Ann. Fac. Sci. Toulouse Math. 17 (2008) 291–308. | DOI | Numdam | MR | Zbl

I. Gentil, A. Guillin and L. Miclo, Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Related Fields 133 (2005) 409–436. | DOI | MR | Zbl

I. Gentil, A. Guillin and L. Miclo, Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoam. 23 (2007) 235–258. | DOI | MR | Zbl

E.D. Gluskin and S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114 (1995) 303–309. | DOI | MR | Zbl

N. Gozlan, Poincaré inequalities and dimension free concentration of measure. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 708–739. | DOI | Numdam | MR | Zbl

L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083. | DOI | MR | Zbl

D.L. Hanson and F.T. Wright, A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42 (1971) 1079–1083. | DOI | MR | Zbl

R. Latała, Tail and moment estimates for sums of independent random vectors with logarithmically concave tails. Studia Math. 118 (1996) 301–304. | DOI | MR | Zbl

R. Latała, Tail and moment estimates for some types of chaos. Studia Math. 135 (1999) 39–53. | DOI | MR | Zbl

R. Latała, Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (2006) 2315–2331. | DOI | MR | Zbl

R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré. In Geometric aspects of functional analysis. Vol. 1745 of Lect. Notes Math. Springer, Berlin (2000) 147–168 | MR | Zbl

M. Ledoux, The concentration of measure phenomenon. Vol. 89 of Mathematical Surveys and Monographs. Amer. Math. Society, Providence, RI (2001). | MR | Zbl

R. Łochowski, Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails. In Approximation and probability. Vol. 72 of Banach Center Publ. Polish Acad. Sci., Warsaw (2006) 161–176. | MR | Zbl

E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009) 1–43. | DOI | MR | Zbl

G. Pisier, Probabilistic methods in the geometry of Banach spaces. In Probability and analysis (Varenna, 1985). Vol. 1206 of Lect. Notes Math. Springer, Berlin (1986) 167–241. | MR | Zbl

O.S. Rothaus, Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64 (1985) 296–313. | DOI | MR | Zbl

O.S. Rothaus, Logarithmic Sobolev inequalities and the growth of $L^p$ norms. Proc. Amer. Math. Soc. 126 (1998) 2309–2314. | DOI | MR | Zbl

J. Shao, Modified logarithmic Sobolev inequalities and transportation cost inequalities in $R^n$. Potential Anal. 31 (2009) 183–202. | DOI | MR | Zbl

A.J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inform. Control 2 (1959) 101–112. | DOI | MR | Zbl

M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon. In Geometric aspects of functional analysis (1989–90). Vol. 1469 of Lect. Notes Math. Springer, Berlin (1991) 94–124. | MR | Zbl

M. Talagrand, The supremum of some canonical processes. Amer. J. Math. 116 (1994) 283–325. | DOI | MR | Zbl