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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Part of this work was done while the second named author was visiting the University of Alberta as a PIMS postdoctoral fellow. The authors would also like to thank the referees for their valuable comments.
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Friedland, O., Giladi, O. & Guédon, O. Small Ball Estimates for Quasi-Norms. J Theor Probab 29, 1624–1643 (2016). https://doi.org/10.1007/s10959-015-0622-z
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DOI: https://doi.org/10.1007/s10959-015-0622-z