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Restricted Khinchine Inequality

Published online by Cambridge University Press:  20 November 2018

Susanna Spektor
Susanna Spektor*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2C1 e-mail: sanaspek@gmail.com
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Abstract

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We prove a Khintchine type inequality under the assumption that the sumof Rademacher randomvariables equals zero. We also showa newtail-bound for a hypergeometric random variable.

Keywords

46B0660E1552A2346B09Khintchine inequalityKahane inequalityRademacher randomvariableshypergeometric distribution
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016 , pp. 204 - 210
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Adamczak, R., Litvak, A. E., Pajor, A.. and Tomczak-Jaegermann, N., Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave ensembles. J. Amer. Math. Soc. 23(2010), no. 2, 535–561. http://dx.doi.Org/10.1090/S0894-0347-09-00650-X Google Scholar
[2] Adamczak, R., Litvak, A. E., Pajor, A.. and Tomczak-Jaegermann, N., Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling. Constr. Approx. 34(2011), no. 1, 61–88. http://dx.doi.Org/10.1007/s00365-010-9117-4 Google Scholar
[3] Adamczak, R., Latala, R.. Litvak, A. E., Pajor, A.. and N. Tomczak-Jaegermann, Tail estimates for norms of sums of log-concave random vectors. Proc. Lond. Math. Soc. 108(2014), no. 3, 600–637. http://dx.doi.Org/10.1112/plms/pdtO31 Google Scholar
[4] Anderson, G. D. and Qiu, S. L., A monotoneity property of the gamma function. Proc. Amer. Math. Soc. 125(1997), no. 11, 3355–3362. http://dx.doi.Org/10.1090/S0002-9939-97-041 52-X Google Scholar
[5] Chafai, D., O. Guédon, G. Lecué, and Pajor, A.. Interaction between compressed sensing, random matrices and high dimensional geometry. Panoramas et Synthèses, 37, Société Mathématique de France, Paris, 2012.Google Scholar
[6] H.|Garling, D. J., Inequalities: a journey into linear analysis. Cambridge University Press, Cambridge, 2007.Google Scholar
[7] Guédon, O., Nayar, P.. and Tkocz, T.. Concentration inequalities and geometry of convex bodies. In: Analytical and Probabilistic Methods in the Geometry of Convex Bodies, IM PAN Lecture Notes, Vol. 2, Warsaw 2014. http://perso-math.univ-mlv.fr/users/guedon.olivier/listepub.html Google Scholar
[8] Johnson, N. L., Kemp, A. W., and Kotz, S.. Univariate discrete distributions. Third éd., Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2005.Google Scholar
[9] Kahane, J.-P., Some random series of functions. Second éd., Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985.Google Scholar
[10] Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces. I. Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Springer-Verlag, Berlin-New York, 1977; Classical Banach spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 97, Springer-Verlag, Berlin-New York, 1979.Google Scholar
[11] Maurey, B., Construction de suites symétriques. C.R. Acad. Sci. Paris Sér. A-B, 288(1979), no. 14, A679-A681.Google Scholar
[12] Milman, V. D. and Schechtman, G.. Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Math., 1200, Springer-Verlag, Berlin, 1986.Google Scholar
[13] O'Rourke, S., A note on the Marchenko-Pastur law for a class of random matrices with dependent entries. Electron. Commun. Probab. 17(2012), no. 28,1-13.Google Scholar
[14] Hitczenko, P., Domination inequality for martingale transforms ofRademacher sequence. Israel J. Math. 84(1993), no. 1-2, 161–178. http://dx.doi.Org/10.1007/BF02761698 Google Scholar
[15] Peskir, G. and Shiryaev, A. N., The inequalities ofKhinchine and expanding sphere of their action. Russian Math. Surveys 50(1995), no. 5, 849–904.Google Scholar
[16] Schechtman, G., Concentration, results and applications. In: Handbook of the geometry of Banach spaces, 2, North-Holland, Amsterdam, 2003. pp. 1603–1634.Google Scholar
[17] Skala, M., Hyper geometric tail inequalities: ending the insanity. arxiv:1311.5939Google Scholar
[18] Spektor, S., Selected topics in asymptotic geometric analysis and approximation theory. Ph.D. Thesis. University of Alberta, 2014.Google Scholar
[19] Tomczak-Jaegermann, N., Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, 38, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1989.Google Scholar