Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities

Journal of Functional Analysis

Volumn 163, Issue 1, 1999, Pages 1-28

  Bobkov, S G ^a,b   Gotze F ^b  

^a SYKTYVKAR STATE UNIVERSITY   (Russian Federation)

^b BIELEFELD UNIVERSITY   (Germany)

Author keywords

Concentration of measure; Exponential integrability; Logarithmic Sobolev inequalities; Transportation inequalities

Indexed keywords

EID: 0002829137     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.1998.3326     Document Type: Article

References (32)

1. Ahlswede R., Gács P., Körner J. Bounds on conditional probabilities with applications in multi-user communication. Z. Wahrsch. Verw. Gebiete. 34:1976;157-177.
2. Aida S., Masuda T., Shigekawa I. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126:1994;83-101.
3. Aida S., Stroock D. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Research Letters. 1:1994;75-86.
4. Bobkov S. G. On Gross' and Talagrand's inequalities on the discrete cube. Vestnik Syktyvkar Univ. 1:1995;12-19.
5. Davies E. B., Simon B. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59:1984;335-395.
6. Diaconis P., Saloff-Coste L. Logarithmic Sobolev inequalities for finite Marlov chains. Ann. Appl. Probab. 6:1996;695-750.
7. Dudley R. M. Real Analysis and Probability. 1989;Pacific Grove.
8. Gross L. Logarithmic Sobolev inequalities. Amer. J. Math. 97:1975;1061-1083.
9. Gross L. Logarithmic Sobolev inequalities and contractivity properties of semigroups, Varenna, 1992. Lecture Notes in Math. 1563:1993;54-88.
10. Harper L. H. Optimal numbering and isopermimetric problems on graphs. J. Combin. Theory. 1:1966;385-393.
11. Kantorovich L. V. K., Akilov G. P. Functional Analysis. 1982;Pergamon, Oxford.
12. M. Ledoux, Isoperimetry and Gaussian analysis, Ecole d 'été de Probabilités de Saint-Flour, 1994, Lecture Notes in Math. 1648, 1996, 165, 294.
13. Ledoux M. Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter. J. Math. Kyoto Univ. 35:1995;211-220.
14. Ledoux M. On Talagrand's deviation inequalities for product measures. ESAIM- Probab. Statist. 1:1996;63-87.
15. Leindler L. On a certain converse of Hölder's inequality. Acta Sci. Math. Szeged. 33:1972;217-223.
16. Levin V. L. The problem of mass transfer in a topological space, and probability measures having given marginal measures on the product of two spaces. Soviet Math. Dokl. 29:1984;638-643.
17. Marton K. A simple proof of the blowing-up lemma. IEEE Trans. Inform. Theory. IT-32:1986;445-446.
18. Marton K. Boundingd. Ann. Probab. 24:1996;857-866.
19. Maurey B. Some deviation inequalities. Geom. Funct. Anal. 1:1991;188-197.
20. Muckenhoupt B. Hardy's inequality with weights. Studia Math. XLIV:1972;31-38.
21. Milman V. D., Schechtman G. Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Math. 1200:1986;Springer-Verlag, Berlin.
22. Pisier G. Probabilistic methods in the geometry of Banach spaces. Probability and Analysis, Varenna, Italy, 1985. Lecture Notes in Math. 1206:1986;Springer-Verlag, Berlin. p. 167-241.
23. Pisier G. The Volume of Convex Bodies and Banach Space Geometry. Tracts in Math. 1989;Cambridge Univ. Press, Cambridge.
24. Précopa A. On logarithmic concave measures and functions. Acta Sci. Math. Szeged. 34:1973;335-343.
25. Rachev S. T. The Monge-Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl. 24:1984;647-671.
26. Rothaus O. S. Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64:1985;296-313.
27. Talagrand M. An isoperimetric theorem on the cube and the Khinchine-Kahane inequalities. Proc. Amer. Math. Soc. 104:1988;905-909.
28. Talagrand M. A new isoperimetric inequality and the concentration of measure phenomenon. Lecture Notes in Math. 1469:1991;94-124.
29. Talagrand M. Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. I.H.E.S. 81:1995;73-205.
30. Talagrand M. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6:1996;587-600.
31. M. Talagrand, Private communication, 1996.
32. Tsirel'son B. S. A geometric approach to a maximum likelyhood estimation for infinite-dimensional Gaussian location, II. Theory Probab. Appl. 30:1985;820-828.