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(np/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.
Citation
Roman Vershynin. "Approximating the moments of marginals of high-dimensional distributions." Ann. Probab. 39 (4) 1591 - 1606, July 2011. https://doi.org/10.1214/10-AOP589
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