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Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles.(English)Zbl 1206.60006

Let \(X\) be a uniformly distributed point in an isotropic convex body in \(\mathbb R^n\), or, more generally, a centered random vector in \(\mathbb R^n\) with a log-concave distribution and identity covariance matrix. Following the lead ofR. Kannan,L. Lovász andM. Simonovits [Random Struct. Algorithms 11, No. 1, 1–50 (1997;Zbl 0895.60075)], several authors have provided quantitative estimates of the distance between the sample covariance matrix, based on \(N\) i.i.d. copies of \(X\), and the identity. In the paper under review, the authors prove that for any \(\varepsilon > 0\), there exists \(C(\varepsilon) > 0\) such that if \(N \geq C(\varepsilon)\cdot n\), then with probability at least \(1-\exp(-c\sqrt{n})\) (\(c > 0\) an absolute constant) this distance is bounded by \(\epsilon\).

MSC:

60B20 Random matrices (probabilistic aspects)
46B09 Probabilistic methods in Banach space theory
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
60E15 Inequalities; stochastic orderings

Cite

References:

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[17]Shahar Mendelson, On weakly bounded empirical processes, Math. Ann. 340 (2008), no. 2, 293 – 314. ·Zbl 1151.60006 ·doi:10.1007/s00208-007-0152-9
[18]Shahar Mendelson and Alain Pajor, On singular values of matrices with independent rows, Bernoulli 12 (2006), no. 5, 761 – 773. ·Zbl 1138.60328 ·doi:10.3150/bj/1161614945
[19]Shahar Mendelson, Alain Pajor, and Nicole Tomczak-Jaegermann, Reconstruction and subgaussian operators in asymptotic geometric analysis, Geom. Funct. Anal. 17 (2007), no. 4, 1248 – 1282. ·Zbl 1163.46008 ·doi:10.1007/s00039-007-0618-7
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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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