Approximating the moments of marginals of high-dimensional distributions

Abstract

For probability distributions on ${\mathbb{R}}^n$, 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.

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)

References in corpus (2)

• Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles
• L_p moments of random vectors via majorizing measures

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