Skip to main content Accessibility help
×
Home
Hostname: page-component-544b6db54f-8tjh8 Total loading time: 0.236 Render date: 2021-10-24T14:39:32.445Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

A Short Proof of Paouris' Inequality

Published online by Cambridge University Press:  20 November 2018

Radosław Adamczak
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail: radamcz@mimuw.edu.pl rlatala@mimuw.edu.pl
Rafał Latała
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail: radamcz@mimuw.edu.pl rlatala@mimuw.edu.pl
Alexander E. Litvak
Affiliation:
Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: alexandr@math.ualberta.ca koles@mimuw.edu.pl
Krzysztof Oleszkiewicz
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail: radamcz@mimuw.edu.pl rlatala@mimuw.edu.pl
Alain Pajor
Affiliation:
Université Paris-Est, Équipe d'Analyse et Mathématiques Appliquées, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France e-mail: nicole.tomczak@ualberta.ca Alain.Pajor@univ-mlv.fr
Nicole Tomczak-Jaegermann
Affiliation:
Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: alexandr@math.ualberta.ca koles@mimuw.edu.pl
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$ ,

$$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$

More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$ ,

$${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle \right|}^{p}})}^{1/p}}.$$

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Adamczak, R., Guédon, O., Latała, R., Litvak, A. E., Oleszkiewicz, K., Pajor, A., and Tomczak-Jaegermann, N., Moment estimates for convex measures. arxiv:1207.6618 CrossRefGoogle Scholar
[2] Adamczak, R., Latała, R., Litvak, A. E., Pajor, A., and Tomczak-Jaegermann, N., Tail estimates fornorms of sums of log-concave random vectors. arxiv:1107.4070. Google Scholar
[3] Adamczak, R., Geometry of log-concave ensembles of random matrices and approximate reconstruction. C. R. Math. Acad. Sci. Paris 349 (2011), no. 1314, 783786. http://dx.doi.org/10.1016/j.crma.2011.06.025 CrossRefGoogle Scholar
[4] Barlow, R. E., Marshall, A.W., and Proschan, F., Properties of probability distributions with monotonehazard rate. Ann. Math. Statist. 34 (1963), 375389. http://dx.doi.org/10.1214/aoms/1177704147 CrossRefGoogle Scholar
[5] Borell, C., Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239252. http://dx.doi.org/10.1007/BF02384761 CrossRefGoogle Scholar
[6] Borell, C., Convex set functions in d-space. Period. Math. Hungar. 6 (1975), no. 2, 111136. http://dx.doi.org/10.1007/BF02018814 CrossRefGoogle Scholar
[7] Borell, C., The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975), no. 2, 207216. http://dx.doi.org/10.1007/BF01425510 CrossRefGoogle Scholar
[8] Davidovic, Ju. S., Korenbljum, B. I., and Hacet, B. I., A certain property of logarithmically concavefunctions. Soviet Math. Dokl. 10 (1969), 447480; translation from Dokl. Akad. Nauk SSSR 185 (1969), 12151218.Google Scholar
[9] Gordon, Y., Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985), no. 4, 265289. http://dx.doi.org/10.1007/BF02759761 CrossRefGoogle Scholar
[10] Kwapień, S. , A remark on the median and the expectation of convex functions of Gaussian vectors. In: Probability in Banach spaces, 9 (Sandjberg, 1993), Progr. Probab., 35, Birkhäuser Boston, Boston, MA, 1994, pp. 271272.Google Scholar
[11] Kwapień, S., Latała, R., and Oleszkiewicz, K., Comparison of moments of sums of independent randomvariables and differential inequalities. J. Funct. Anal. 136 (1996), no. 1, 258268. http://dx.doi.org/10.1006/jfan.1996.0030 CrossRefGoogle Scholar
[12] Lifshits, M. A., Gaussian random functions. Mathematics and its Applications, 322, Kluwer Academic Publishers, Dordrecht, 1995.Google Scholar
[13] Litvak, A. E., Milman, V. D., and Schechtman, G., Averages of norms and quasi-norms. Math. Ann. 312 (1998), no. 1, 95124. http://dx.doi.org/10.1007/s002080050213 CrossRefGoogle Scholar
[14] Paouris, G., Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), no. 5, 10211049. http://dx.doi.org/10.1007/s00039-006-0584-5 CrossRefGoogle Scholar
[15] Sudakov, V. N. and Cirel’son, B. S., Extremal properties of half-spaces for spherically invariantmeasures. J. Sov. Math. 9 (1978), 918; translation from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 41 (1974), 1424, 165.CrossRefGoogle Scholar
You have Access
13
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A Short Proof of Paouris' Inequality
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A Short Proof of Paouris' Inequality
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A Short Proof of Paouris' Inequality
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *