The Annals of Probability

Weighted Poincaré-type inequalities for Cauchy and other convex measures

Sergey G. Bobkov and Michel Ledoux

Full-text: Open access

Abstract

Brascamp–Lieb-type, weighted Poincaré-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κ-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 403-427.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099916

Digital Object Identifier
doi:10.1214/08-AOP407

Mathematical Reviews number (MathSciNet)
MR2510011

Zentralblatt MATH identifier
1178.46041

Subjects
Primary: 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12] 60B11: Probability theory on linear topological spaces [See also 28C20] 60G07: General theory of processes

Keywords
Brascamp–Lieb-type inequalities weighted Poincaré-type inequalities logarithmic Sovolev inequalities infimum convolution measure concentration Cheeger-type inequalities

Citation

Bobkov, Sergey G.; Ledoux, Michel. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 (2009), no. 2, 403--427. doi:10.1214/08-AOP407. https://projecteuclid.org/euclid.aop/1241099916


Export citation

References

  • [1] Aida, S. and Stroock, D. (1994). Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1 75–86.
  • [2] Blanchet, A., Bonforte, M., Dolbeault, J., Grillo, G. and Vazquez, J.-L. (2007). Hardy–Poincaré inequalities and applications to nonlinear diffusions. C. R. Math. Acad. Sci. Paris 344 431–436.
  • [3] Blanchet, A., Bonforte, M., Dolbeault, J., Grillo, G. and Vazquez, J.-L. (2009). Asymptotics of the fast diffusion equation via entropy estimates. Arch. Ration. Mech. Anal. To appear.
  • [4] Bobkov, S. G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903–1921.
  • [5] Bobkov, S. G. (2007). Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12 1072–1100 (electronic).
  • [6] Bobkov, S. G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80 669–696.
  • [7] Bobkov, S. G. and Ledoux, M. (2000). From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 1028–1052.
  • [8] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
  • [9] Borell, C. (1975). Convex set functions in d-space. Period. Math. Hungar. 6 111–136.
  • [10] Brascamp, H. J. and Lieb, E. H. (1976). 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. Funct. Anal. 22 366–389.
  • [11] Das Gupta, S. (1980). Brunn–Minkowski inequality and its aftermath. J. Multivariate Anal. 10 296–318.
  • [12] Denzler, J. and McCann, R. J. (2005). Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology. Arch. Ration. Mech. Anal. 175 301–342.
  • [13] Gardner, R. J. (2002). The Brunn–Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 355–405 (electronic).
  • [14] Gromov, M. and Milman, V. D. (1983). A topological application of the isoperimetric inequality. Amer. J. Math. 105 843–854.
  • [15] Kannan, R., Lovász, L. and Simonovits, M. (1995). Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 541–559.
  • [16] Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. In Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 120–216. Springer, Berlin.
  • [17] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [18] Leindler, L. (1972). On a certain converse of Hölder’s inequality. II. Acta Sci. Math. (Szeged) 33 217–223.
  • [19] Lovász, L. and Simonovits, M. (1993). Random walks in a convex body and an improved volume algorithm. Random Structures Algorithms 4 359–412.
  • [20] Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal. 1 188–197.
  • [21] Muckenhoupt, B. (1972). Hardy’s inequality with weights. Studia Math. 44 31–38.
  • [22] Prékopa, A. (1971). Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32 301–316.
  • [23] Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34 335–343.
  • [24] Rothaus, O. S. (1985). Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities. J. Funct. Anal. 64 296–313.
  • [25] Tsirelson, B. S. (1985). A geometric approach to maximum likelihood estimation for an infinite-dimensional Gaussian location. II. Teor. Veroyatnost. i Primenen. 30 772–779.