Small Ball Estimates for Quasi-Norms |
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Authors: | Omer Friedland Ohad Giladi Olivier Guédon |
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Institution: | 1.Institut de Mathématiques de Jussieu,Université Pierre et Marie Curie,Paris,France;2.School of Mathematical and Physical Sciences,University of Newcastle,Callaghan,Australia;3.Laboratoire d’Analyse et Mathématiques Appliquées,Université Paris-Est,Marne-la-Vallée,France |
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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). |
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