Concentration of measure and logarithmic Sobolev inequalities

 Access to full text

 Full (PDF)

1. [Ai] S. Aida. Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal.158, 152-185 (1998). Zbl0914.47041MR1641566
2. [A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal.126, 83-101 (1994). Zbl0846.46020MR1305064
3. [A-S] S. Aida, D. Stroock. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett.1, 75-86 (1994). Zbl0862.60064MR1258492
4. [Al] N. Alon. Eigenvalues and expanders. J. Combin. Theory, Ser. B, 38, 78-88 (1987). MR875835
5. [A-L] C. Ané, M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Preprint (1998). Zbl0964.60063MR1757600
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. Zbl0856.47026MR1307413
7. [Ba2] D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in Stochastic Analysis. 43-75 (1997). World Scientific. MR1654503
8. [Ba-E] D. Bakry, M. Emery. Diffusions hypercontractives. Séminaire de Probabilites XIX. Lecture Notes in Math.1123, 177-206 (1985). Springer-Verlag. Zbl0561.60080MR889476
9. [Ba-L] D. Bakry, M. Ledoux. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent. math.123, 259-281 (1996). Zbl0855.58011MR1374200
10. [B-L-Q] D. Bakry, M. Ledoux, Z. Qian. Preprint (1997). 
11. [Be] W. Beckner. Personal communication (1998). 
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. Series284, 107-121 (1993). Longman. Zbl0801.60019MR1354164
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. Zbl0920.62042MR1462939
14. [B-M2] L. Birgé, P. Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence (1998). Bernoulli, to appear. Zbl0954.62033MR1653272
15. [B-B-M] A. Barren, L. Birgé, P. Massart. Risk bounds for model selection via penalization (1998). Probab. Theory Relat. Fields, to appear. Zbl0946.62036MR1679028
16. [Bob1] S. Bobkov. and On Gross' Talagrand's inequalities on the discrete cube. Vestnik of Syktyvkar University, Ser. 1, 1, 12-19 (1995) (in Russian). Zbl1011.60500MR1717705
17. [Bob2] S. Bobkov. Some extremal properties of Bernoulli distribution. Probability Theor. Appl.41, 877-884 (1996). Zbl0895.60012MR1687168
18. [Bob3] S. Bobkov. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal.135, 39-49 (1996). Zbl0838.60013MR1367623
19. [Bob4] S. Bobkov. An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probability25, 206-214 (1997). Zbl0883.60031MR1428506
20. [Bob5] S. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures (1998). Ann. Probability, to appear. Zbl0964.60013MR1742893
21. [B-G] S. Bobkov, F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities (1997). J. Funct. Anal., to appear. Zbl0924.46027MR1682772
22. [B-H] S. Bobkov, C. Houdré. Isoperimetric constants for product probability measures. Ann. Probability25, 184-205 (1997). Zbl0878.60013MR1428505
23. [B-L1] S. Bobkov, M. Ledoux. Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential measure. Probab. Theory Relat. Fields107, 383-400 (1997). Zbl0878.60014MR1440138
24. [B-L2] S. Bobkov, M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal.156, 347-365 (1998). Zbl0920.60002MR1636948
25. [Bon] A. Bonami. Etude des coefficients de Fourier des fonctions de LP(G). Ann. Inst. Fourier20, 335-402 (1970). Zbl0195.42501MR283496
26. [Bor] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. math.30, 207-216 (1975). Zbl0292.60004MR399402
27. [Br] R. Brooks. On the spectrum of non-compact manifolds with finite volume. Math. Z.187, 425-437 (1984). MR757481
28. [Cha1] I. Chavel. Eigenvalues in Riemannian geometry. Academic Press (1984). Zbl0551.53001MR768584
29. [Cha2] I. Chavel. Riemannian geometry - A modern introduction. Cambridge Univ. Press (1993). Zbl0810.53001MR1271141
30. [Che] S.-Y. Cheng. Eigenvalue comparison theorems and its geometric applications. Math. Z.143, 289-297 (1975). Zbl0329.53035MR378001
31. [Da] E.B. Davies. Heat kernel and spectral theory. Cambridge Univ. Press (1989). MR990239
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). Zbl0568.47034MR766493
33. [De] A. Dembo. Information inequalities and concentration of measure. Ann. Probability25, 927-939 (1997). Zbl0880.60018MR1434131
34. [D-Z] A. Dembo, O. Zeitouni. Transportation approach to some concentration inequalities in product spaces. Elect. Comm. in Probab.1, 83-90 (1996). Zbl0916.28003MR1423908
35. [De-S] J.-D. Deuschel, D. Stroock. Large deviations. Academic Press (1989). Zbl0705.60029MR997938
36. [D-SC] P. Diaconis, L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob.6, 695-750 (1996). Zbl0867.60043MR1410112
37. [Eh] A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand.53, 281-301 (1983). Zbl0542.60003MR745081
38. [G-M] M. Gromov, V.D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math.105, 843-854 (1983). Zbl0522.53039MR708367
39. [Gr1] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math.97, 1061-1083 (1975). Zbl0318.46049MR420249
40. [Gr2] L. Gross. Logarithmic Sobolev inequalities and contractive properties of semigroups. Dirichlet Forms, Varenna1992. Lect. Notes in Math.1563, 54-88 (1993). Springer-Verlag. Zbl0812.47037MR1292277
41. [G-R] L. Gross, O. Rothaus. Herbst inequalities for supercontractive semigroups. Preprint (1997). Zbl0928.47032MR1648283
42. [H-Y] Y. Higuchi, N. Yoshida. Analytic conditions and phase transition for Ising models. Lecture Notes in Japanese (1995). 
43. [H-S] R. Holley, D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys.46, 1159-1194 (1987). Zbl0682.60109MR893137
44. [H-T] C. Houdré, P. Tetali. Concentration of measure for products of Markov kernels via functional inequalities. Preprint (1997). Zbl0986.28004MR1827807
45. [Hs1] E.P. Hsu. Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Commun. Math. Phys.189, 9-16 (1997). Zbl0892.58083MR1478528
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). Zbl1069.60500MR1678311
47. [J-S] W.B. Johnson, G. Schechtman. Remarks on Talagrand's deviation inequality for Rademacher functions. Longhorn Notes, Texas (1987). Zbl0753.60024
48. [Kl] C.A.J. Klaassen. On an inequality of Chernoff. Ann. Probability13, 966-974 (1985). Zbl0576.60015MR799431
49. [K-S] A. Korzeniowski, D. Stroock. An example in the theory of hypercontractive semigroups. Proc. Amer. Math. Soc.94, 87-90 (1985). Zbl0577.47043MR781062
50. [Kw-S] S. Kwapien, J. Szulga. Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces. Ann. Probability19, 369-379 (1991). Zbl0718.60044MR1085342
51. [K-L-O] S. Kwapien, R. Latala, K. Oleszkiewicz. Comparison of moments of sums of independent random variables and differential inequalities. J. Funct. Anal.136, 258-268 (1996). Zbl0864.60014MR1375162
52. [Le1] M. Ledoux. Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci.Paris, 306, 79-92 (1988). Zbl0646.60006MR929114
53. [Le2] M. Ledoux. Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter. J. Math. Kyoto Univ.35, 211-220 (1995). Zbl0836.60074MR1346225
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. Zbl0874.60005MR1600888
55. [Le4] M. Ledoux. On Talagrand's deviation inequalities for product measures. ESAIM Prob. & Stat.1, 63-87 (1996). Zbl0869.60013
56. [L-T] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991). Zbl0748.60004MR1102015
57. [L-Y] T.Y. Lee, H.-T. Yau. Logarithmic Sobolev inequality fo some models of random walks. Preprint (1998). MR1675008
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). Zbl0429.35054MR551166
59. [Ly] T. Lyons. Random thoughts on reversible potential theory. Summer School in Potentiel Theory, Joensuu1990. Publications in Sciences26, 71-114University of Joensuu. Zbl0757.31007MR1175217
60. [MD] C. McDiarmid. On the method of bounded differences. Surveys in Combinatorics. London Math. Soc. Lecture Notes141, 148-188 (1989). Cambridge Univ. Press. Zbl0712.05012MR1036755
61. [Mar1] K. Marton. Bounding d-distance by information divergence: a method to prove measure concentration. Ann. Probability24, 857-866 (1996). Zbl0865.60017MR1404531
62. [Mar2] K. Marton. A measure concentration inequality for contracting Markov chains. Geometric and Funct. Anal.6, 556-571 (1997). Zbl0856.60072MR1392329
63. [Mar3] K. Marton. Measure concentration for a class of random processes. Probab. Theory Relat. Fields110, 427-439 (1998). Zbl0927.60050MR1616492
64. [Mar4] K. Marton. On a measure concentration of Talagrand for dependent random variables. Preprint (1998). MR1392329
65. [Mas] P. Massart. About the constants in Talagrand's deviation inequalities for empirical processes (1998). Ann. Probability, to appear. Zbl1140.60310MR1782276
66. [Mau1] B. Maurey. Constructions de suites symétriques. C. R. Acad. Sci.Paris288, 679-681 (1979). Zbl0398.46019MR533901
67. [Mau2] B. Maurey. Some deviations inequalities. Geometric and Funct. Anal.1, 188-197 (1991). Zbl0756.60018MR1097258
68. [Mi] V.D. Milman. Dvoretzky theorem - Thirty years later. Geometric and Funct. Anal.2, 455-479 (1992) . Zbl0787.46016MR1191569
69. [M-S] V.D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200 (1986). Springer-Verlag. Zbl0606.46013MR856576
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). Zbl0506.46022MR674060
71. [O-V] F. Otto, C. Villani. Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality. Preprint (1998). MR1760620
72. [Pi] M.S. Pinsker. Information and information stability of random variables and processes. Holden-Day, San Franscico (1964). Zbl0125.09202MR213190
73. [Ro1] O. Rothaus. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal.42, 358-367 (1981). Zbl0471.58027MR620581
74. [Ro2] O. Rothaus. Hypercontractivity and the Bakry-Emery criterion for compact Lie groups. J. Funct. Anal.65, 358-367 (1986). Zbl0589.58036MR826433
75. [Ro3] O. Rothaus. Logarithmic Sobolev inequalities and the growth of Lp norms (1996). Zbl0896.46021
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). Zbl0816.53027MR1292948
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. Zbl0885.60061MR1490046
78. [Sa] P.-M. Samson. Concentration of measure inequalities for Markov chains and Φ-mixing processes. Preprint (1998). MR1756011
79. [Sc] M. Schmuckenschläger. Martingales, Poincaré type inequalities and deviations inequalities. J. Funct. Anal.155, 303-323 (1998). Zbl0924.58111MR1624545
80. [St] D. Stroock. Logarithmic Sobolev inequalities for Gibbs states. Dirichlet forms, Varenna 1992. Lecture Notes in Math.1563, 194-228 (1993). Zbl0801.60056MR1292280
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). Zbl0794.46025MR1153990
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). Zbl0395.28007MR365680
83. [Tak] M. Takeda. On a martingale method for symmetric diffusion process and its applications. Osaka J. Math.26, 605-623 (1989). Zbl0717.60090MR1021434
84. [Ta1] M. Talagrand. An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc.104, 905-909 (1988). Zbl0691.60015MR964871
85. [Ta2] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability17, 1546-1570 (1989). Zbl0692.60016MR1048946
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. Zbl0818.46047MR1122615
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. Zbl0791.60004MR1159286
88. [Ta5] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probability22, 28-76 (1994). Zbl0798.60051MR1258865
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). Zbl0864.60013MR1361756
90. [Ta7] M. Talagrand. A new look at independence. Ann. Probability, 24, 1-34 (1996). Zbl0858.60019MR1387624
91. [Ta8] M. Talagrand. New concentration inequalities in product spaces. Invent. math.126, 505-563 (1996). Zbl0893.60001MR1419006
92. [Ta9] M. Talagrand. Transportation cost for Gaussian and other product measures. Geometric and Funct. Anal.6, 587-600 (1996). Zbl0859.46030MR1392331
93. [Wan] F.-Y. Wang. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Proab. Theory Relat. Fields109, 417-424 (1997). Zbl0887.35012MR1481127
94. [Wat] G.N. Watson. A treatise on the theory of Bessel functions. Cambridge Univ. Press (1944). Zbl0063.08184MR10746
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). Zbl0561.53046MR794292

1. Guangfei Li, Yu Miao, Huiming Peng, Liming Wu, Poincaré and log-Sobolev inequality for stationary Gaussian processes and moving average processes
2. Aldéric Joulin, Nicolas Privault, Functional inequalities for discrete gradients and application to the geometric distribution
3. Aldéric Joulin, Nicolas Privault, Functional inequalities for discrete gradients and application to the geometric distribution
4. H. Djellout, A. Guillin, L. Wu, Moderate deviations of empirical periodogram and non-linear functionals of moving average processes
5. Liming Wu, A deviation inequality for non-reversible Markov processes
6. H. Djellout, A. Guillin, Large and moderate deviations for moving average processes
7. Laurent Miclo, About projections of logarithmic Sobolev inequalities
8. Djalil Chafaï, Gaussian maximum of entropy and reversed log-Sobolev inequality
9. Franck Barthe, Neil O'Connell, Matchings and the variance of Lipschitz functions
10. Lorenzo Bertini, Nicoletta Cancrini, Filippo Cesi, The spectral gap for a Glauber-type dynamics in a continuous gas