Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 13 (2016) open journal systems 

Hyperbolic Measures on Infinite Dimensional Spaces

Sergey G. Bobkov, University of Minnesota
James C. Melbourne, University of Minnesota

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

AMS 2000 subject classifications: 60B11, 28C20, 60F10.

Keywords: Hyperbolic (convex) measures, dimension, localization, dilation of sets.

Creative Common LOGO

Full Text: PDF

Bobkov, Sergey G., Melbourne, James C., Hyperbolic Measures on Infinite Dimensional Spaces, Probability Surveys, 13, (2016), 57-88 (electronic). DOI: 10.1214/14-PS238.


[B1]     Bobkov, S. G. Remarks on the growth of Lp-norms of polynomials. Geometric aspects of functional analysis, 27–35, Lecture Notes in Math., 1745, Springer, Berlin, 2000. MR1796711

[B2]     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. MR1968725

[B3]     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. MR2001838

[B4]     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. MR2035184

[B5]     Bobkov, S. G. Large deviations and isoperimetry over convex probability measures. Electron. J. Probab. 12 (2007), 1072–1100. MR2336600

[B6]     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. MR2347041

[B-M]     Bobkov, S. G., Madiman, M. Concentration of the information in data with log-concave distributions. Ann. Probab. 39 (2011), no. 4, 1528–1543. MR2857249

[B-N]     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. MR2742901

[BN]     Barndorff-Nielsen, O. Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5 (1978), no. 3, 151–157 MR0509451

[Bog]     Bogachev, V. I. Measure Theory. Vol. I, II. Springer-Verlag, Berlin, 2007. Vol. I: xviii+500 pp., Vol. II: xiv+575 pp. MR2267655

[B-S-S]     Bogachev, V. I., Smolyanov, O. G., Sobolev, V. I. Topological vector spaces and their applications (Russian). Moscow, Izhevsk, 2012, 584 pp.

[Bor1]     Borell, C. Convex measures on locally convex spaces. Ark. Math. 12 (1974), 239–252. MR0388475

[Bor2]     Borell, C. Convex set functions in d-space. Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR0404559

[Bor3]     Borell, Christer. Convexity of measures in certain convex cones in vector space  sigma-algebras. Mathematica Scandinavica 53 (1983), 125–144. MR0733944

[B-L]     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. MR0450480

[B-Z]     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. MR0936419

[D-K-H]     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. MR0241584

[F]     Fradelizi, M. Concentration inequalities for s-concave measures of dilations of Borel sets and applications. Electron. J. Probab. 14 (2009), no. 71, 2068–2090. MR2550293

[F-G1]     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. MR2060645

[F-G2]     Fradelizi, M., Guédon, O. A generalized localization theorem and geometric inequalities for convex bodies. Adv. Math. 204 (2006), no. 2, 509–529. MR2249622

[G-M]     Gromov, M., Milman, V. D. Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces. Composition Math. 62 (1987), 263–282. MR0901393

[G]     Guédon, O. Kahane-Khinchine type inequalities for negative exponent. Mathematika 46 (1999), no. 1, 165–173. MR1750653

[H-O]     Hadwiger, H., Ohmann, D. Brunn-Minkowskischer Satz und Isoperimetrie. Math. Z., 66 (1956), 1–8. MR0082697

[I]     Ibragimov, I. A. On the composition of unimodal distributions. (Russian) Teor. Veroyatnost. i Primenen. 1 (1956), 283–288. MR0087249

[K-L-S]     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

[K-A]     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. MR0788496

[K-S]     Kotz, Samuel, and Saralees Nadarajah. Multivariate t-distributions and their applications. Cambridge University Press, 2004. MR2038227

[L-T]     Ledoux, M.,Talagrand, M. Probability in Banach Spaces: isoperimetry and processes. Vol. 23. Springer, 1991. MR2814399

[L-S]     Lovász, L. Simonovits, M. Random walks in a convex body and an improved volume algorithm. Random Structures Algor. 4 (1993), no. 4, 359–412. MR1238906

[M]     Meyer, P.-A. Probability and potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966 xiii+266 pp. MR0205288

[N-S-V]     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. MR1925887

[P-S]     Puig, P., Stephens, M, A. Goodness-of-fit tests for the hyperbolic distribution. Canad. J. Statist. 29 (2001), no. 2, 309–320. MR1840711

[P-W]     Payne, L. E., Weinberger, H. F. An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960), 286–292. MR0117419

[Ph]     Phelps, R. R. Lectures on Choquet’s theorem. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966 v+130 pp. MR1835574

[Pr]     Prékopa, A. Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32 (1971), 301–316. MR0315079

[R]     Rudin, W. Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973, xiii+397 pp. MR1157815

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787