Approximating the moments of marginals of high-dimensional distributions

2009/11/30 by Roman Vershynin
Mathematics · #math.PR #math.FA

paper · pdf · doi:10.1214/10-aop589

published as Annals of Probability 2011, Vol. 39, No. 4, 1591--1606 · Published in at http://dx.doi.org/10.1214/10-AOP589 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

arxiv created 2012/11/16 · arxiv updated 2014/05/21

Abstract

For probability distributions on ℝⁿ, we study the optimal sample size N = N(n,p) that suffices to uniformly approximate the pth moments of all one-dimensional marginals. Under the assumption that the marginals have bounded 4p moments, we obtain the optimal bound N=O(n^p/2) for p > 2. This bound goes in the direction of bridging the two recent results: a theorem of Guedon and Rudelson [Adv. Math. 208 (2007) 798-823] which has an extra logarithmic factor in the sample size, and a result of Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535-561] which requires stronger subexponential moment assumptions.

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