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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 39 open journal systems 


Large deviations and isoperimetry over convex probability measures with heavy tails

Sergey G Bobkov, University of Minnesota


Abstract
Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn-Minkowski-type


Full text: PDF

Pages: 1072-1100

Published on: August 25, 2007
Bibliography
  1. Borell, Christer. Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239--252. MR0388475 (52 #9311)
  2. Borell, C. Convex set functions in $d$-space. Period. Math. Hungar. 6 (1975), no. 2, 111--136. MR0404559 (53 #8359)
  3. Aida, Shigeki. Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 (1998), no. 1, 152--185. MR1641566 (2000d:60125)
  4. Barthe, F. Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12 (2002), no. 1, 32--55. MR1904555 (2003d:28017)
  5. Barthe, F.; Cattiaux, P.; Roberto, C. Concentration for independent random variables with heavy tails. AMRX Appl. Math. Res. Express 2005, no. 2, 39--60. MR2173316 (2006h:60031)
  6. Bertini, Lorenzo; Zegarlinski, Bogusl aw. Coercive inequalities for Gibbs measures. J. Funct. Anal. 162 (1999), no. 2, 257--286. MR1682059 (2000c:60158)
  7. Bobkov, S. Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24 (1996), no. 1, 35--48. MR1387625 (97e:60027)
  8. Bobkov, S. G. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999), no. 4, 1903--1921. MR1742893 (2001h:60026)
  9. Bobkov, Sergey G. Large deviations via transference plans. Advances in mathematics research, Vol. 2, 151--175, Adv. Math. Res., 2, Nova Sci. Publ., Hauppauge, NY, 2003. MR2035184 (2005b:60061)
  10. Bobkov, S. G.; Houdré, C. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997), no. 1, 184--205. MR1428505 (98g:60032)
  11. Bourgain, J. On the distribution of polynomials on high-dimensional convex sets. Geometric aspects of functional analysis (1989--90), 127--137, Lecture Notes in Math., 1469, Springer, Berlin, 1991. MR1122617 (92j:52007)
  12. Brascamp, Herm Jan; Lieb, Elliott H. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Functional Analysis 22 (1976), no. 4, 366--389. MR0450480 (56 #8774)
  13. Dyer, Martin; Frieze, Alan. Computing the volume of convex bodies: a case where randomness provably helps. Probabilistic combinatorics and its applications (San Francisco, CA, 1991), 123--169, Proc. Sympos. Appl. Math., 44, Amer. Math. Soc., Providence, RI, 1991. MR1141926 (93a:52004)
  14. Gromov, M.; Milman, V. D. Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Compositio Math. 62 (1987), no. 3, 263--282. MR0901393 (89f:46031)
  15. Guédon, Olivier. Kahane-Khinchine type inequalities for negative exponent. Mathematika 46 (1999), no. 1, 165--173. MR1750653 (2001d:26037)
  16. Kannan, R.; Lovász, L.; Simonovits, M. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995), no. 3-4, 541--559. MR1318794 (96e:52018)
  17. Knothe, Herbert. Contributions to the theory of convex bodies. Michigan Math. J. 4 (1957), 39--52. MR0083759 (18,757b)
  18. Latal a, Rafal. On the equivalence between geometric and arithmetic means for log-concave measures. Convex geometric analysis (Berkeley, CA, 1996), 123--127, Math. Sci. Res. Inst. Publ., 34, Cambridge Univ. Press, Cambridge, 1999. MR1665584 (2000a:60025)
  19. Ledoux, Michel. Isoperimetry and Gaussian analysis. Lectures on probability theory and statistics (Saint-Flour, 1994), 165--294, Lecture Notes in Math., 1648, Springer, Berlin, 1996. MR1600888 (99h:60002)
  20. Ledoux, Michel. Spectral gap, logarithmic Sobolev constant, and geometric bounds. Surveys in differential geometry. Vol. IX, 219--240, Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004. MR2195409 (2007f:58049)
  21. Liggett, Thomas M. $Lsb 2$ rates of convergence for attractive reversible nearest particle systems: the critical case. Ann. Probab. 19 (1991), no. 3, 935--959. MR1112402 (93a:60159)
  22. Lovász, László; Simonovits, Miklós. The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume. 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), 346--354, IEEE Comput. Soc. Press, Los Alamitos, CA, 1990. MR1150706 (93e:68035)
  23. Lovász, L.; Simonovits, M. Random walks in a convex body and an improved volume algorithm. Random Structures Algorithms 4 (1993), no. 4, 359--412. MR1238906 (94m:90091)
  24. Mathieu, P. Quand l'inégalité log-Sobolev implique l'inégalité de trou spectral. (French) [When the log-Sobolev inequality implies the spectral gap inequality] Séminaire de Probabilités, XXXII, 30--35, Lecture Notes in Math., 1686, Springer, Berlin, 1998. MR1651227 (2000a:60143)
  25. Nazarov, F.; Sodin, M.; Volcprime berg, A. The geometric Kannan-Lovász-Simonovits lemma, dimension-free estimates for the distribution of the values of polynomials, and the distribution of the zeros of random analytic functions. (Russian) Algebra i Analiz 14 (2002), no. 2, 214--234; translation in St. Petersburg Math. J. 14 (2003), no. 2, 351--366 MR1925887 (2004e:60086)
  26. Payne, L. E.; Weinberger, H. F. An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 1960 286--292 (1960). MR0117419 (22 #8198)
  27. Röckner, Michael; Wang, Feng-Yu. Weak Poincaré inequalities and $Lsp 2$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001), no. 2, 564--603. MR1856277 (2002j:47075)
















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Electronic Journal of Probability. ISSN: 1083-6489