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On generic chaining and the smallest singular value of random matrices with heavy tails.(English)Zbl 1242.60008

Summary: We present a very general chaining method which allows one to control the supremum of the empirical process \(\sup_{h\in H}|N^{-1}\sum^N_{i=1}h^2(X_i)-\mathbb{E} h^2|\) in rather general situations. We use this method to establish two main results. First, a quantitative (non-asymptotic) version of the celebrated Bai-Yin theorem on the singular values of a random matrix with i.i.d. entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when \(H=\{\langle t,\cdot\rangle:t\in T\}\), \(T\subset\mathbb{R}^n\) and \(\mu\) is an isotropic, unconditional, log-concave measure.

MSC:

60B20 Random matrices (probabilistic aspects)
60G05 Foundations of stochastic processes

Cite

References:

[1]R. Adamczak, R. Latała, A. Litvak, A. Pajor, N. Tomczak-Jaegermann, Chevet type inequality and norms of submatrices, preprint.; R. Adamczak, R. Latała, A. Litvak, A. Pajor, N. Tomczak-Jaegermann, Chevet type inequality and norms of submatrices, preprint. ·Zbl 1253.60005
[2]Adamczak, R.; Litvak, A.; Pajor, A.; Tomczak-Jaegermann, N., Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, J. Amer. Math. Soc., 23, 535-561 (2010) ·Zbl 1206.60006
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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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