Sample covariance matrices of heavy-tailed distributions
By Konstantin E. Tikhomirov. 2016
56 citations recorded by the source corpus.
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 $$K^{-1}\|\Sigma_N-I_n\|_{2\to 2}\leq\frac{1}{N}\max\limits_{i\leq N}\|X_i\|^2 +\Bigl(\frac{n}{N}\Bigr)^{1-2/p}\log^4\frac{N}{n}+\Bigl(\frac{n}{N}\Bigr)^{1-2/\min(p,4)}$$ with probability at least $1-\frac{1}{n}$, where $K>0$ depends only on $B$ and $p$. In particular, for all $p>4$ we obtain a quantitative Bai-Yin type theorem.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/d1b7beebf77e0414a63db2636385c6f3d9cbc61d
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices (2019, background): “A series of work obtained control on these quantities under weaker assumptions [1, 50, 63, 69].”