Concentration of measure and logarithmic Sobolev inequalities

  • Michel Ledoux
Cours Spécialisés
Part of the Lecture Notes in Mathematics book series (LNM, volume 1709)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ai] S. Aida. Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158, 152–185 (1998).MathSciNetCrossRefMATHGoogle Scholar
  2. [A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126, 83–101 (1994).MathSciNetCrossRefMATHGoogle Scholar
  3. [A-S] S. Aida, D. Stroock. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1, 75–86 (1994).MathSciNetCrossRefMATHGoogle Scholar
  4. [Al] N. Alon. Eigenvalues and expanders. J. Combin. Theory, Ser. B, 38, 78–88 (1987).Google Scholar
  5. [A-L] C. Ané, M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Preprint (1998).Google Scholar
  6. [Ba1] D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. Ecole d'Eté de Probabilités de St-Flour. Lecture Notes in Math. 1581, 1–114 (1994). Springer-Verlag.MathSciNetCrossRefGoogle Scholar
  7. [Ba2] D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in Stochastic Analysis. 43–75 (1997). World Scientific.Google Scholar
  8. [Ba-E] D. Bakry, M. Emery. Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 177–206 (1985). Springer-Verlag.MathSciNetCrossRefMATHGoogle Scholar
  9. [Ba-L] D. Bakry, M. Ledoux. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent. math. 123, 259–281 (1996).MathSciNetMATHGoogle Scholar
  10. [B-L-Q] D. Bakry, M. Ledoux, Z. Qian. Preprint (1997).Google Scholar
  11. [Be] W. Beckner. Personal communication (1998).Google Scholar
  12. [BA-L] G. Ben Arous, M. Ledoux. Schilder's large deviation principle without topology. Asymptotic problems in probability theory: Wiener functionals and asymptotics. Pitman Research Notes in Math. Series 284, 107–121 (1993). Longman.MATHGoogle Scholar
  13. [B-M1] L. Birgé, P. Massart. From model selection to adaptive estimation. Festschrift for Lucien LeCam: Research papers in Probability and Statistics (D. Pollard, E. Torgersen and G. Yang, eds.) 55–87 (1997). Springer-Verlag.Google Scholar
  14. [B-M2] L. Birgé, P. Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence (1998). Bernoulli, to appear.Google Scholar
  15. [B-B-M] A. Barron, L. Birgé, P. Massart. Risk bounds for model selection via penalization (1998). Probab. Theory Relat. Fields, to appear.Google Scholar
  16. [Bob1] S. Bobkov. On Gross' and Talagrand's inequalities on the discrete cube. Vestnik of Syktyvkar University, Ser. 1, 1, 12–19 (1995) (in Russian).MathSciNetMATHGoogle Scholar
  17. [Bob2] S. Bobkov. Some extremal properties of Bernoulli distribution. Probability Theor. Appl. 41, 877–884 (1996).MathSciNetMATHGoogle Scholar
  18. [Bob3] S. Bobkov. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135, 39–49 (1996).MathSciNetCrossRefMATHGoogle Scholar
  19. [Bob4] S. Bobkov. An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probability 25, 206–214 (1997).MathSciNetCrossRefMATHGoogle Scholar
  20. [Bob5] S. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures (1998). Ann. Probability, to appear.Google Scholar
  21. [B-G] S. Bobkov, F., Götze. Exponential integrability and transporation cost related to logarithmic Sobolev inequalities (1997). J. Funct. Anal., to appear.Google Scholar
  22. [B-H] S. Bobkov, C. Houdré. Isoperimetric constants for product probability measures. Ann. Probability 25, 184–205 (1997).MathSciNetCrossRefMATHGoogle Scholar
  23. [B-L1] S. Bobkov, M. Ledoux. Poincarés inequalities and Talagrand's concentration phenomenon for the exponential measure. Probab. Theory Relat. Fields 107, 383–400 (1997).MathSciNetCrossRefMATHGoogle Scholar
  24. [B-L2] S. Bobkov, M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156, 347–365 (1998).MathSciNetCrossRefMATHGoogle Scholar
  25. [Bon] A. Bonami. Etude des coefficients de Fourier des fonctions de L p(G). Ann. Inst. Fourier 20, 335–402 (1970).MathSciNetCrossRefMATHGoogle Scholar
  26. [Bor] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. math. 30, 207–216 (1975).MathSciNetCrossRefMATHGoogle Scholar
  27. [Br] R. Brooks. On the spectrum of non-compact manifolds with finite volume. Math. Z. 187, 425–437 (1984).MathSciNetCrossRefMATHGoogle Scholar
  28. [Cha1] I. Chavel. Eigenvalues in Riemannian geometry. Academic Press (1984). *** DIRECT SUPPORT *** A00I6C60 00003Google Scholar
  29. [Cha2] I. Chavel. Riemannian geometry—A modern introduction. Cambridge Univ. Press (1993).Google Scholar
  30. [Che] S.-Y. Cheng. Eigenvalue comparison theorems and its geometric applications. Math. Z. 143, 289–297 (1975).MathSciNetCrossRefMATHGoogle Scholar
  31. [Da] E. B. Davies. Heat kernel and spectral theory. Cambridge Univ. Press (1989).Google Scholar
  32. [D-S] E. B. Davies, B. Simon. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335–395 (1984).MathSciNetCrossRefMATHGoogle Scholar
  33. [De] A. Dembo. Information inequalities and concentration of measure. Ann. Probability 25, 927–939 (1997).MathSciNetCrossRefMATHGoogle Scholar
  34. [D-Z] A. Dembo, O. Zeitouni. Transportation approach to some concentration inequalities in product spaces. Elect. Comm. in Probab. 1, 83–90 (1996).MathSciNetCrossRefMATHGoogle Scholar
  35. [De-S] J.-D. Deuschel, D. Stroock. Large deviations. Academic Press (1989).Google Scholar
  36. [D-SC] P. Diaconis, L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob. 6, 695–750 (1996).MathSciNetCrossRefMATHGoogle Scholar
  37. [Eh] A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand. 53, 281–301 (1983).MathSciNetMATHGoogle Scholar
  38. [G-M] M. Gromov, V. D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843–854 (1983).MathSciNetCrossRefMATHGoogle Scholar
  39. [Gr1] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975).MathSciNetCrossRefMATHGoogle Scholar
  40. [Gr2] L. Gross. Logarithmic Sobolev inequalities and contractive properties of semigroups. Dirichlet Forms, Varenna 1992. Lect. Notes in Math. 1563, 54–88 (1993). Springer-Verlag.CrossRefGoogle Scholar
  41. [G-R] L. Gross, O. Rothaus. Herbst inequalities for supercontractive semigroups. Preprint (1997).Google Scholar
  42. [H-Y] Y. Higuchi, N. Yoshida. Analytic conditions and phase transition for Ising models. Lecture Notes in Japanese (1995).Google Scholar
  43. [H-S] R. Holley, D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46, 1159–1194 (1987).MathSciNetCrossRefMATHGoogle Scholar
  44. [H-T] C. Houdré, P. Tetali. Concentration of measure for products of Markov kernels via functional inequalities. Preprint (1997).Google Scholar
  45. [Hs1] E. P. Hsu. Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Commun. Math. Phys. 189, 9–16 (1997).MathSciNetCrossRefMATHGoogle Scholar
  46. [Hs2] E. P. Hsu. Analysis on Path and Loop Spaces (1996). To appear in IAS/Park City Mathematics Series, Vol. 5, edited by E. P. Hsu and S. R. S. Varadhan, American Mathematical Society and Institute for Advanced Study (1997).Google Scholar
  47. [J-S] W. B. Johnson, G. Schechtman. Remarks on Talagrand's deviation inequality for Rademacher functions. Longhorn Notes, Texas (1987).Google Scholar
  48. [Kl] C. A. J. Klaassen. On an inequality of Chernoff. Ann. Probability 13, 966–974 (1985).MathSciNetCrossRefMATHGoogle Scholar
  49. [K-S] A. Korzeniowski, D. Stroock. An example in the theory of hypercontractive semigroups. Proc. Amer. Math. Soc. 94, 87–90 (1985).MathSciNetCrossRefMATHGoogle Scholar
  50. [Kw-S] S. Kwapień, J. Szulga. Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces. Ann. Probability 19, 369–379 (1991).MathSciNetCrossRefMATHGoogle Scholar
  51. [K-L-O] S. Kwapień, R. Latala, K. Oleszkiewicz. Comparison of moments of sums of independent random variables and differential inequalities. J. Funct. Anal. 136, 258–268 (1996).MathSciNetCrossRefMATHGoogle Scholar
  52. [Le1] M. Ledoux. Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci. Paris, 306, 79–92 (1988).MathSciNetMATHGoogle Scholar
  53. [Le2] M. Ledoux. Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter. J. Math. Kyoto Univ. 35, 211–220 (1995).MathSciNetMATHGoogle Scholar
  54. [Le3] M. Ledoux. Isoperimetry and Gaussian Analysis. Ecole d'Eté de Probabilités de St-Flour 1994. Lecture Notes in Math. 1648, 165–294 (1996). Springer-Verlag.MathSciNetCrossRefMATHGoogle Scholar
  55. [Le4] M. Ledoux. On Talagrand's deviation inequalities for product measures. ESAIM Prob. & Stat. 1, 63–87 (1996).MathSciNetCrossRefMATHGoogle Scholar
  56. [L-T] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991).Google Scholar
  57. [L-Y] T. Y. Lee, H.-T. Yau. Logarithmic Sobolev inequality fo some models of random walks. Preprint (1998).Google Scholar
  58. [Li] P. Li. A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana Univ. Math. J. 28, 1013–1019 (1979).MathSciNetCrossRefMATHGoogle Scholar
  59. [Ly] T. Lyons. Random thoughts on reversible potential theory: Summer School in Potentiel Theory, Joensuu 1990. Publications in Sciences 26, 71–114 University of Joensuu.Google Scholar
  60. [MD] C. McDiarmid. On the method of bounded differences. Surveys in Combinatorics. London Math. Soc. Lecture Notes 141, 148–188 (1989). Cambridge Univ. Press.MathSciNetMATHGoogle Scholar
  61. [Mar1] K. Marton. Bounding Open image in new window-distance by information divergence: a method to prove measure concentration. Ann. Probability 24, 857–866 (1996).MathSciNetCrossRefMATHGoogle Scholar
  62. [Mar2] K. Marton. A measure concentration inequality for contracting Markov chains. Geometric and Funct. Anal. 6, 556–571 (1997).MathSciNetCrossRefMATHGoogle Scholar
  63. [Mar3] K. Marton. Measure concentration for a class of random processes. Probab. Theory Relat. Fields 110, 427–439 (1998).MathSciNetCrossRefMATHGoogle Scholar
  64. [Mar4] K. Marton. On a measure concentration of Talagrand for dependent random variables. Preprint (1998).Google Scholar
  65. [Mas] P. Massart. About the constants in Talagrand's deviation inequalities for empirical processes (1998). Ann. Probability, to appear.Google Scholar
  66. [Mau1] B. Maurey. Constructions de suites symétriques. C. R. Acad. Sci. Paris 288, 679–681 (1979).MathSciNetMATHGoogle Scholar
  67. [Mau2] B. Maurey. Some deviations inequalities. Geometric and Funct. Anal. 1, 188–197 (1991).MathSciNetCrossRefMATHGoogle Scholar
  68. [Mi] V. D. Milman. Dvoretzky theorem-Thirty years later. Geometric and Funct. Anal. 2, 455–479 (1992).MathSciNetCrossRefMATHGoogle Scholar
  69. [M-S] V. D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200 (1986). Springer-Verlag.Google Scholar
  70. [M-W] C. Muller, F. Weissler. Hypercontractivity of the heat semigroup for ultraspherical polynomials and on the n-sphere. J. Funct. Anal. 48, 252–283 (1982).MathSciNetCrossRefMATHGoogle Scholar
  71. [O-V] F. Otto, C. Villani. Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality. Preprint (1998).Google Scholar
  72. [Pi] M. S. Pinsker. Information and information stability of random variables and processes. Holden-Day, San Franscico (1964).MATHGoogle Scholar
  73. [Ro1] O. Rothaus. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42, 358–367 (1981).MathSciNetMATHGoogle Scholar
  74. [Ro2] O. Rothaus. Hypercontractivity and the Bakry-Emery criterion for compact Lie groups. J. Funct. Anal. 65, 358–367 (1986).MathSciNetCrossRefMATHGoogle Scholar
  75. [Ro3] O. Rothaus. Logarithmic Sobolev inequalities and the growth of L p norms (1996).Google Scholar
  76. [SC1] L. Saloff-Coste. Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below. Colloquium Math. 67, 109–121 (1994).MathSciNetMATHGoogle Scholar
  77. [SC2] L. Saloff-Coste. Lectures on finite Markov chains. Ecole d'Eté de Probabilités de St-Flour 1996. Lecture Notes in Math. 1665, 301–413 (1997). Springer-Verlag.MathSciNetCrossRefMATHGoogle Scholar
  78. [Sa] P.-M. Samson. Concentration of measure inequalities for Markov chains and ϕ-mixing processes. Preprint (1998).Google Scholar
  79. [Sc] M. Schmuckenschläger. Martingales, Poincaré type inequalities and deviations inequalities. J. Funct. Anal. 155, 303–323 (1998).MathSciNetCrossRefMATHGoogle Scholar
  80. [St] D. Stroock. Logarithmic Sobolev inequalities for Gibbs states. Dirichlet forms, Varenna 1992. Lecture Notes in Math. 1563, 194–228 (1993).MathSciNetCrossRefMATHGoogle Scholar
  81. [S-Z] D. Stroock, B. Zegarlinski. The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal. 104, 299–326 (1992).MathSciNetCrossRefMATHGoogle Scholar
  82. [S-T] V. N. Sudakov, B. S. Tsirel'son. Extremal properties of half-spaces for spherically invariant measures. J. Soviet. Math. 9, 9–18 (1978); translated from Zap. Nauch. Sem. L.O.M.I. 41, 14–24 (1974).CrossRefMATHGoogle Scholar
  83. [Tak] M. Takeda. On a martingale method for symmetric diffusion process and its applications. Osaka J. Math. 26, 605–623 (1989).MathSciNetMATHGoogle Scholar
  84. [Ta1] M. Talagrand. An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc., 104, 905–909 (1988).MathSciNetCrossRefMATHGoogle Scholar
  85. [Ta2] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability 17, 1546–1570 (1989).MathSciNetCrossRefMATHGoogle Scholar
  86. [Ta3] M. Talagrand. A new isoperimetric inequality for product measure and the concentration of measure phenomenon. Israel Seminar (GAFA), Lecture Notes in Math. 1469, 91–124 (1991). Springer-Verlag.MathSciNetGoogle Scholar
  87. [Ta4] M. Talagrand. Some isoperimetric inequalities and their applications. Proc. of the International Congress of Mathematicians, Kyoto 1990, vol. II, 1011–1024 (1992). Springer-Verlag.MathSciNetMATHGoogle Scholar
  88. [Ta5] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probability 22, 28–76 (1994).MathSciNetCrossRefMATHGoogle Scholar
  89. [Ta6] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l'I.H.E.S. 81, 73–205 (1995).MathSciNetCrossRefMATHGoogle Scholar
  90. [Ta7] M. Talagrand. A new look at independence. Ann. Probability, 24, 1–34 (1996).MathSciNetCrossRefMATHGoogle Scholar
  91. [Ta8] M. Talagrand. New concentration inequalities in product spaces. Invent. math. 126, 505–563 (1996).MathSciNetCrossRefMATHGoogle Scholar
  92. [Ta9] M. Talagrand. Transportation cost for Gaussian and other product measures. Geometric and Funct. Anal. 6, 587–600 (1996).MathSciNetCrossRefMATHGoogle Scholar
  93. [Wan] F.-Y. Wang. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Proab. Theory Relat. Fields 109, 417–424 (1997).MathSciNetCrossRefMATHGoogle Scholar
  94. [Wat] G. N. Watson. A treatise on the theory of Bessel functions. Cambridge Univ. Press (1944).Google Scholar
  95. [Z-Y] J. Q. Zhong, H. C. Yang. On the estimate of the first eigenvalue of a compact Riemanian manifold. Sci. Sinica Ser. A 27 (12), 1265–1273 (1984). *** DIRECT SUPPORT *** A00I6C60 00004MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Michel Ledoux

There are no affiliations available

Personalised recommendations