Probability Surveys

Hyperbolic measures on infinite dimensional spaces

Abstract

Localization and dilation procedures are discussed for infinite dimensional $\alpha$-concave measures on abstract locally convex spaces (following Borell’s hierarchy of hyperbolic measures).

Article information

Source
Probab. Surveys, Volume 13 (2016), 57-88.

Dates
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ps/1465321421

Digital Object Identifier
doi:10.1214/14-PS238

Mathematical Reviews number (MathSciNet)
MR3511344

Zentralblatt MATH identifier
1342.60006

Citation

Bobkov, Sergey G.; Melbourne, James. Hyperbolic measures on infinite dimensional spaces. Probab. Surveys 13 (2016), 57--88. doi:10.1214/14-PS238. https://projecteuclid.org/euclid.ps/1465321421

References

• [1] Bobkov, S. G. Remarks on the growth of $L^{p}$-norms of polynomials. Geometric aspects of functional analysis, 27–35, Lecture Notes in Math., 1745, Springer, Berlin, 2000.
• [2] Bobkov, S. G. Some generalizations of Prokhorov’s results on Khinchin-type inequalities for polynomials. (Russian) Teor. Veroyatnost. i Primenen. 45 (2000), no. 4, 745–748. Translation in: Theory Probab. Appl. 45 (2002), no. 4, 644–647.
• [3] Bobkov, S. G. Localization proof of the isoperimetric Bakry-Ledoux inequality and some applications. Teor. Veroyatnost. i Primenen. 47 (2002), no. 2, 340–346. Translation in: Theory Probab. Appl. 47 (2003), no. 2, 308–314.
• [4] Bobkov, S. G. Large deviations via transference plans. Advances in mathematics research, Vol. 2, 151–175, Adv. Math. Res., 2, Nova Sci. Publ., Hauppauge, NY, 2003.
• [5] Bobkov, S. G. Large deviations and isoperimetry over convex probability measures. Electron. J. Probab. 12 (2007), 1072–1100.
• [6] Bobkov, S. G. On isoperimetric constants for log-concave probability distributions. Geometric aspects of functional analysis, 81–88, Lecture Notes in Math., 1910, Springer, Berlin, 2007.
• [7] Bobkov, S. G., Madiman, M. Concentration of the information in data with log-concave distributions. Ann. Probab. 39 (2011), no. 4, 1528–1543.
• [8] Bobkov, S. G., Nazarov, F. L. Sharp dilation-type inequalities with fixed parameter of convexity. J. Math. Sci. (N.Y.) 152 (2008), no. 6, 826–839. Translation from: Zap. Nauchn. Sem. POMI 351 (2007), Veroyatnost i Statistika, 12, 54–78.
• [9] Barndorff-Nielsen, O. Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5 (1978), no. 3, 151–157
• [10] Bogachev, V. I. Measure Theory. Vol. I, II. Springer-Verlag, Berlin, 2007. Vol. I: xviii+500 pp., Vol. II: xiv+575 pp.
• [11] Bogachev, V. I., Smolyanov, O. G., Sobolev, V. I. Topological vector spaces and their applications (Russian). Moscow, Izhevsk, 2012, 584 pp.
• [12] Borell, C. Convex measures on locally convex spaces. Ark. Math. 12 (1974), 239–252.
• [13] Borell, C. Convex set functions in $d$-space. Period. Math. Hungar. 6 (1975), no. 2, 111–136.
• [14] Borell, Christer. Convexity of measures in certain convex cones in vector space $\ sigma$-algebras. Mathematica Scandinavica 53 (1983), 125–144.
• [15] Brascamp, H. J., Lieb, E. 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. Funct. Anal. 22 (1976), no. 4, 366–389.
• [16] Burago Yu. D., Zalgaller, V. A. Geometric inequalities. Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskii, Springer Series in Soviet Mathematics, xiv+331 pp.
• [17] Davidovich, Ju. S., Korenbljum, B. I., Hacet, B. I. A certain property of logarithmically concave functions. (Russian) Dokl. Akad. Nauk SSSR 185 (1969), 1215–1218.
• [18] Fradelizi, M. Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications. Electron. J. Probab. 14 (2009), no. 71, 2068–2090.
• [19] Fradelizi, M., Guédon, O. The extreme points of subsets of s -concave probabilities and a geometric localization theorem. Discrete Comput. Geom. 31 (2004), no. 2, 327–335.
• [20] Fradelizi, M., Guédon, O. A generalized localization theorem and geometric inequalities for convex bodies. Adv. Math. 204 (2006), no. 2, 509–529.
• [21] Gromov, M., Milman, V. D. Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Composition Math. 62 (1987), 263–282.
• [22] Guédon, O. Kahane-Khinchine type inequalities for negative exponent. Mathematika 46 (1999), no. 1, 165–173.
• [23] Hadwiger, H., Ohmann, D. Brunn-Minkowskischer Satz und Isoperimetrie. Math. Z., 66 (1956), 1–8.
• [24] Ibragimov, I. A. On the composition of unimodal distributions. (Russian) Teor. Veroyatnost. i Primenen. 1 (1956), 283–288.
• [25] 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.
• [26] Kantorovich, L. V., Akilov, G. P. Functional Analysis. Translated from the Russian by Howard L. Silcock. Second edition. Pergamon Press, Oxford-Elmsford, N.Y., 1982. xiv+589 pp.
• [27] Kotz, Samuel, and Saralees Nadarajah. Multivariate t-distributions and their applications. Cambridge University Press, 2004.
• [28] Ledoux, M.,Talagrand, M. Probability in Banach Spaces: isoperimetry and processes. Vol. 23. Springer, 1991.
• [29] Lovász, L. Simonovits, M. Random walks in a convex body and an improved volume algorithm. Random Structures Algor. 4 (1993), no. 4, 359–412.
• [30] Meyer, P.-A. Probability and potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966 xiii+266 pp.
• [31] Nazarov, F., Sodin, M., Vol’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.
• [32] Puig, P., Stephens, M, A. Goodness-of-fit tests for the hyperbolic distribution. Canad. J. Statist. 29 (2001), no. 2, 309–320.
• [33] Payne, L. E., Weinberger, H. F. An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960), 286–292.
• [34] Phelps, R. R. Lectures on Choquet’s theorem. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966 v+130 pp.
• [35] Prékopa, A. Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32 (1971), 301–316.
• [36] Rudin, W. Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973, xiii+397 pp.