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Concentration of measure and logarithmic Sobolev inequalities

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Séminaire de Probabilités XXXIII

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References

  • [Ai] S. Aida. Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158, 152–185 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  • [A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126, 83–101 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [A-S] S. Aida, D. Stroock. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1, 75–86 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Al] N. Alon. Eigenvalues and expanders. J. Combin. Theory, Ser. B, 38, 78–88 (1987).

    Google Scholar 

  • [A-L] C. Ané, M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Preprint (1998).

    Google Scholar 

  • [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.

    Article  MathSciNet  Google Scholar 

  • [Ba2] D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in Stochastic Analysis. 43–75 (1997). World Scientific.

    Google Scholar 

  • [Ba-E] D. Bakry, M. Emery. Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 177–206 (1985). Springer-Verlag.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ba-L] D. Bakry, M. Ledoux. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent. math. 123, 259–281 (1996).

    MathSciNet  MATH  Google Scholar 

  • [B-L-Q] D. Bakry, M. Ledoux, Z. Qian. Preprint (1997).

    Google Scholar 

  • [Be] W. Beckner. Personal communication (1998).

    Google Scholar 

  • [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.

    MATH  Google Scholar 

  • [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 

  • [B-M2] L. Birgé, P. Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence (1998). Bernoulli, to appear.

    Google Scholar 

  • [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 

  • [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).

    MathSciNet  MATH  Google Scholar 

  • [Bob2] S. Bobkov. Some extremal properties of Bernoulli distribution. Probability Theor. Appl. 41, 877–884 (1996).

    MathSciNet  MATH  Google Scholar 

  • [Bob3] S. Bobkov. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135, 39–49 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bob5] S. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures (1998). Ann. Probability, to appear.

    Google Scholar 

  • [B-G] S. Bobkov, F., Götze. Exponential integrability and transporation cost related to logarithmic Sobolev inequalities (1997). J. Funct. Anal., to appear.

    Google Scholar 

  • [B-H] S. Bobkov, C. Houdré. Isoperimetric constants for product probability measures. Ann. Probability 25, 184–205 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [B-L2] S. Bobkov, M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156, 347–365 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bon] A. Bonami. Etude des coefficients de Fourier des fonctions de L p(G). Ann. Inst. Fourier 20, 335–402 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  • [Bor] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. math. 30, 207–216 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [Br] R. Brooks. On the spectrum of non-compact manifolds with finite volume. Math. Z. 187, 425–437 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  • [Cha1] I. Chavel. Eigenvalues in Riemannian geometry. Academic Press (1984). *** DIRECT SUPPORT *** A00I6C60 00003

    Google Scholar 

  • [Cha2] I. Chavel. Riemannian geometry—A modern introduction. Cambridge Univ. Press (1993).

    Google Scholar 

  • [Che] S.-Y. Cheng. Eigenvalue comparison theorems and its geometric applications. Math. Z. 143, 289–297 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [Da] E. B. Davies. Heat kernel and spectral theory. Cambridge Univ. Press (1989).

    Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [De] A. Dembo. Information inequalities and concentration of measure. Ann. Probability 25, 927–939 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [D-Z] A. Dembo, O. Zeitouni. Transportation approach to some concentration inequalities in product spaces. Elect. Comm. in Probab. 1, 83–90 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [De-S] J.-D. Deuschel, D. Stroock. Large deviations. Academic Press (1989).

    Google Scholar 

  • [D-SC] P. Diaconis, L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob. 6, 695–750 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Eh] A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand. 53, 281–301 (1983).

    MathSciNet  MATH  Google Scholar 

  • [G-M] M. Gromov, V. D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843–854 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  • [Gr1] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • [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.

    Article  Google Scholar 

  • [G-R] L. Gross, O. Rothaus. Herbst inequalities for supercontractive semigroups. Preprint (1997).

    Google Scholar 

  • [H-Y] Y. Higuchi, N. Yoshida. Analytic conditions and phase transition for Ising models. Lecture Notes in Japanese (1995).

    Google Scholar 

  • [H-S] R. Holley, D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46, 1159–1194 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  • [H-T] C. Houdré, P. Tetali. Concentration of measure for products of Markov kernels via functional inequalities. Preprint (1997).

    Google Scholar 

  • [Hs1] E. P. Hsu. Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Commun. Math. Phys. 189, 9–16 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [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 

  • [J-S] W. B. Johnson, G. Schechtman. Remarks on Talagrand's deviation inequality for Rademacher functions. Longhorn Notes, Texas (1987).

    Google Scholar 

  • [Kl] C. A. J. Klaassen. On an inequality of Chernoff. Ann. Probability 13, 966–974 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [K-S] A. Korzeniowski, D. Stroock. An example in the theory of hypercontractive semigroups. Proc. Amer. Math. Soc. 94, 87–90 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [Le1] M. Ledoux. Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci. Paris, 306, 79–92 (1988).

    MathSciNet  MATH  Google Scholar 

  • [Le2] M. Ledoux. Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter. J. Math. Kyoto Univ. 35, 211–220 (1995).

    MathSciNet  MATH  Google Scholar 

  • [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.

    Article  MathSciNet  MATH  Google Scholar 

  • [Le4] M. Ledoux. On Talagrand's deviation inequalities for product measures. ESAIM Prob. & Stat. 1, 63–87 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [L-T] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991).

    Google Scholar 

  • [L-Y] T. Y. Lee, H.-T. Yau. Logarithmic Sobolev inequality fo some models of random walks. Preprint (1998).

    Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [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 

  • [MD] C. McDiarmid. On the method of bounded differences. Surveys in Combinatorics. London Math. Soc. Lecture Notes 141, 148–188 (1989). Cambridge Univ. Press.

    MathSciNet  MATH  Google Scholar 

  • [Mar1] K. Marton. Bounding -distance by information divergence: a method to prove measure concentration. Ann. Probability 24, 857–866 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Mar2] K. Marton. A measure concentration inequality for contracting Markov chains. Geometric and Funct. Anal. 6, 556–571 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [Mar3] K. Marton. Measure concentration for a class of random processes. Probab. Theory Relat. Fields 110, 427–439 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  • [Mar4] K. Marton. On a measure concentration of Talagrand for dependent random variables. Preprint (1998).

    Google Scholar 

  • [Mas] P. Massart. About the constants in Talagrand's deviation inequalities for empirical processes (1998). Ann. Probability, to appear.

    Google Scholar 

  • [Mau1] B. Maurey. Constructions de suites symétriques. C. R. Acad. Sci. Paris 288, 679–681 (1979).

    MathSciNet  MATH  Google Scholar 

  • [Mau2] B. Maurey. Some deviations inequalities. Geometric and Funct. Anal. 1, 188–197 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [Mi] V. D. Milman. Dvoretzky theorem-Thirty years later. Geometric and Funct. Anal. 2, 455–479 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [M-S] V. D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200 (1986). Springer-Verlag.

    Google Scholar 

  • [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).

    Article  MathSciNet  MATH  Google Scholar 

  • [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 

  • [Pi] M. S. Pinsker. Information and information stability of random variables and processes. Holden-Day, San Franscico (1964).

    MATH  Google Scholar 

  • [Ro1] O. Rothaus. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42, 358–367 (1981).

    MathSciNet  MATH  Google Scholar 

  • [Ro2] O. Rothaus. Hypercontractivity and the Bakry-Emery criterion for compact Lie groups. J. Funct. Anal. 65, 358–367 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ro3] O. Rothaus. Logarithmic Sobolev inequalities and the growth of L p norms (1996).

    Google Scholar 

  • [SC1] L. Saloff-Coste. Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below. Colloquium Math. 67, 109–121 (1994).

    MathSciNet  MATH  Google Scholar 

  • [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.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sa] P.-M. Samson. Concentration of measure inequalities for Markov chains and ϕ-mixing processes. Preprint (1998).

    Google Scholar 

  • [Sc] M. Schmuckenschläger. Martingales, Poincaré type inequalities and deviations inequalities. J. Funct. Anal. 155, 303–323 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  • [St] D. Stroock. Logarithmic Sobolev inequalities for Gibbs states. Dirichlet forms, Varenna 1992. Lecture Notes in Math. 1563, 194–228 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  • [S-Z] D. Stroock, B. Zegarlinski. The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal. 104, 299–326 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  • [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).

    Article  MATH  Google Scholar 

  • [Tak] M. Takeda. On a martingale method for symmetric diffusion process and its applications. Osaka J. Math. 26, 605–623 (1989).

    MathSciNet  MATH  Google Scholar 

  • [Ta1] M. Talagrand. An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc., 104, 905–909 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta2] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability 17, 1546–1570 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  • [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.

    MathSciNet  Google Scholar 

  • [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.

    MathSciNet  MATH  Google Scholar 

  • [Ta5] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probability 22, 28–76 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta6] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l'I.H.E.S. 81, 73–205 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta7] M. Talagrand. A new look at independence. Ann. Probability, 24, 1–34 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta8] M. Talagrand. New concentration inequalities in product spaces. Invent. math. 126, 505–563 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Ta9] M. Talagrand. Transportation cost for Gaussian and other product measures. Geometric and Funct. Anal. 6, 587–600 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  • [Wan] F.-Y. Wang. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Proab. Theory Relat. Fields 109, 417–424 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  • [Wat] G. N. Watson. A treatise on the theory of Bessel functions. Cambridge Univ. Press (1944).

    Google Scholar 

  • [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 00004

    MathSciNet  MATH  Google Scholar 

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Jacques Azéma Michel Émery Michel Ledoux Marc Yor

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Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol 1709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096511

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