L_p moments of random vectors via majorizing measures

By Olivier Guédon, Mark Rudelson. 2005

55 citations recorded by the source corpus.

For a random vector X in R^n, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of L_p. We also prove that for an isotropic log-concave random vector, we only need about n^{p/2} \log n sample points so that the empirical p-th moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ff6a13ce221a55569817baa8f8d6da442d962947

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In Applied Mathematics

In Numerical Analysis

In Statistics and Probability

Citations in the corpus

Concentration of mass on convex bodies (2006, background): “See also [20] for an extension of this result.”
Conditional quantile processes based on series or many regressors (2011, methodology): “ax α∈Sm−1 En (α′Zi)2 ∨E (α′Zi)2 . (G.39) The factor φn will be used to bound the quantities ǫ 0 and ǫ 1 in the analysis for the rate of convergence. Next we state a result due to Gu´edon and Rudelson [18] specialized to our framework. Theorem 13 (Gu´edon and Rudelson [18]). Let Zi ∈ Rm, i= 1,...,n, ”
Some new asymptotic theory for least squares series: Pointwise and uniform results (2012, methodology): “A version of this inequality was derived by Guédon and Rudelson (2007) using generalized entropy (majorizing measure) arguments.”
Empirical Processes with a Bounded Ψ1 Diameter (2010, background): “[21, 9, 15, 32, 20, 30, 25, 4]), in which the estimate has been improved from the initial N ≥ c(ε, δ)n2 in [21] to the best possible estimate of N ≥ c(ε, δ)n, proved in [1].”