# Sample covariance matrices of heavy-tailed distributions

@article{Tikhomirov2016SampleCM,
title={Sample covariance matrices of heavy-tailed distributions},
author={Konstantin E. Tikhomirov},
journal={arXiv: Probability},
year={2016}
}
Let $p>2$, $B\geq 1$, $N\geq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $\sup\limits_{a\in S^{n-1}}{\mathrm E}|\langle X,a\rangle|^p\leq B$. Further, let $X_1,X_2,\dots,X_N$ be independent copies of $X$, and $\Sigma_N:=\frac{1}{N}\sum_{i=1}^N X_i {X_i}^T$ be the sample covariance matrix. We prove that $$K^{-1}\|\Sigma_N-I_n\|_{2\to 2}\leq\frac{1}{N}\max\limits_{i\leq N}\|X_i\|^2 +\Bigl(\frac{n}{N}\Bigr)^{1-2/p}\log^4\frac{N}{n}+\Bigl… 21 Citations Approximating the covariance ellipsoid It is shown that if the slabs are replaced by randomly generated ellipsoids defined using X, the same degree of approximation is true when N \geq c_1d\eta^{-2}\log(2/\eta). The smallest singular value of a shifted d-regular random square matrix • Mathematics Probability Theory and Related Fields • 2018 We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let$$C_1<d< c n/\log ^2
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