Electronic Journal of Probability

Estimates of norms of log-concave random matrices with dependent entries

Marta Strzelecka

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We prove estimates for $\mathbb{E} \| X: \ell _{p'}^{n} \to \ell _{q}^{m}\|$ for $p,q\ge 2$ and any random matrix $X$ having the entries of the form $a_{ij}Y_{ij}$, where $Y=(Y_{ij})_{1\le i\le m, 1\le j\le n}$ has i.i.d. isotropic log-concave rows and $p'$ denotes the Hölder conjugate of $p$. This generalises a result of Guédon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide an analogous bound for $m\times n$ random matrices, whose entries form an unconditional vector in $\mathbb{R} ^{mn}$. We also prove bounds for norms of matrices whose entries are certain Gaussian mixtures.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 107, 15 pp.

Received: 3 April 2019
Accepted: 18 September 2019
First available in Project Euclid: 2 October 2019

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Digital Object Identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 15B52: Random matrices

random matrices operator norm log-concave vectors unconditional vectors

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Strzelecka, Marta. Estimates of norms of log-concave random matrices with dependent entries. Electron. J. Probab. 24 (2019), paper no. 107, 15 pp. doi:10.1214/19-EJP365. https://projecteuclid.org/euclid.ejp/1569981822

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