Let , , and let be a random vector in with independent K–subgaussian components. We show that for every 1–Lipschitz convex function f in (the Lipschitzness with respect to the Euclidean metric), ,
where is a universal constant. The estimates are optimal in the sense that for every and there exist a product probability distribution X in with K–subgaussian components, and a 1–Lipschitz convex function f, with
The obtained deviation estimates for subgaussian variables are in sharp contrast with the case of variables with bounded –quasi-norms for .
References
R. Adamczak, Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 2, 221–238. MR2163396R. Adamczak, Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 2, 221–238. MR2163396
D. Bakry and M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), no. 2, 259–281. MR1374200D. Bakry and M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), no. 2, 259–281. MR1374200
S. G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 (1997), no. 1, 206–214. MR1428506S. G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 (1997), no. 1, 206–214. MR1428506
E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), no. 2, 335–395. MR0766493E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), no. 2, 335–395. MR0766493
N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, Ann. Probab. 37 (2009), no. 6, 2480–2498. MR2573565N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, Ann. Probab. 37 (2009), no. 6, 2480–2498. MR2573565
N. Gozlan, C. Roberto and P.-M. Samson, From dimension free concentration to the Poincaré inequality, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 899–925. MR3311918N. Gozlan, C. Roberto and P.-M. Samson, From dimension free concentration to the Poincaré inequality, Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 899–925. MR3311918
N. Gozlan, C. Roberto, P.-M. Samson, P. Tetali, Kantorovich duality for general transport costs and applications, J. Funct. Anal. 273 (2017), no. 11, 3327–3405. MR3706606N. Gozlan, C. Roberto, P.-M. Samson, P. Tetali, Kantorovich duality for general transport costs and applications, J. Funct. Anal. 273 (2017), no. 11, 3327–3405. MR3706606
N. Gozlan, C. Roberto, P.-M. Samson, Y. Shu, P. Tetali, Characterization of a class of weak transport-entropy inequalities on the line, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 3, 1667–1693. MR3825894N. Gozlan, C. Roberto, P.-M. Samson, Y. Shu, P. Tetali, Characterization of a class of weak transport-entropy inequalities on the line, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 3, 1667–1693. MR3825894
Y. Klochkov and N. Zhivotovskiy, Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method, Electron. J. Probab. 25 (2020), Paper No. 22, 30 pp. MR4073683Y. Klochkov and N. Zhivotovskiy, Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method, Electron. J. Probab. 25 (2020), Paper No. 22, 30 pp. MR4073683
R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis, 147–168, Lecture Notes in Math., 1745, Springer, Berlin. MR1796718R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis, 147–168, Lecture Notes in Math., 1745, Springer, Berlin. MR1796718
M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89, American Mathematical Society, Providence, RI, 2001. MR1849347M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89, American Mathematical Society, Providence, RI, 2001. MR1849347
V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, 1200, Springer-Verlag, Berlin, 1986. MR0856576V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, 1200, Springer-Verlag, Berlin, 1986. MR0856576
F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361–400. MR1760620F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361–400. MR1760620
Y. Shu and M. Strzelecki, A characterization of a class of convex log-Sobolev inequalities on the real line, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 4, 2075–2091. MR3865667Y. Shu and M. Strzelecki, A characterization of a class of convex log-Sobolev inequalities on the real line, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 4, 2075–2091. MR3865667
V. N. Sudakov and B. S. Tsirelson, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24, 165. MR0365680V. N. Sudakov and B. S. Tsirelson, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24, 165. MR0365680
M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 73–205. MR1361756M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 73–205. MR1361756
R. van Handel, Probability in High Dimension, Lecture notes, https://web.math.princeton.edu/rvan/APC550.pdfR. van Handel, Probability in High Dimension, Lecture notes, https://web.math.princeton.edu/rvan/APC550.pdf