Sample covariance matrices of heavy-tailed distributions
Sample covariance matrices of heavy-tailed distributions
复制标题
重尾分布的样本协方差矩阵
DOI：
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发表时间：
2016
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影响因子：
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通讯作者：
K. Tikhomirov
中科院分区：
文献类型：
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作者：
K. Tikhomirov
Let $p>2$, $Bgeq 1$, $Ngeq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $suplimits_{ain S^{n-1}}{mathrm E}|langle X,a
angle|^pleq B$. 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 $$K^{-1}|Sigma_N-I_n|_{2 o 2}leqfrac{1}{N}maxlimits_{ileq N}|X_i|^2 +Bigl(frac{n}{N}Bigr)^{1-2/p}log^4frac{N}{n}+Bigl(frac{n}{N}Bigr)^{1-2/min(p,4)}$$ 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.
Let $p>2$, $Bgeq 1$, $Ngeq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $suplimits_{ain S^{n-1}}{mathrm E}|langle X,a
angle|^pleq B$. 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 $$K^{-1}|Sigma_N-I_n|_{2 o 2}leqfrac{1}{N}maxlimits_{ileq N}|X_i|^2 +Bigl(frac{n}{N}Bigr)^{1-2/p}log^4frac{N}{n}+Bigl(frac{n}{N}Bigr)^{1-2/min(p,4)}$$ 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.