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Journal of Theoretical Probability

, Volume 29, Issue 4, pp 1624–1643 | Cite as

Small Ball Estimates for Quasi-Norms

  • Omer Friedland
  • Ohad GiladiEmail author
  • Olivier Guédon
Article

Abstract

This note contains two types of small ball estimates for random vectors in finite-dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood–Offord type estimates for quasi-norms. This generalizes results which were previously obtained in Friedland and Sodin (C R Math Acad Sci Paris 345(9):513–518, 2007), and Rudelson and Vershynin (Commun Pure Appl Math 62(12):1707–1739, 2009).

Keywords

Small ball estimates Sobolev norm Littlewood–Offord type estimates 

Mathematics Subject Classification (2010)

60D05 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParisFrance
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  3. 3.Laboratoire d’Analyse et Mathématiques AppliquéesUniversité Paris-EstMarne-la-ValléeFrance

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