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Uzu Lim
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ThisA variant of this (taking $\frac1m$ instead of $\frac1{m-1}$ for the empirical covariance) is answered on Proposition 2.6, page 10 of https://arxiv.org/pdf/2110.06357.pdf .

This is answered on Proposition 2.6, page 10 of https://arxiv.org/pdf/2110.06357.pdf .

A variant of this (taking $\frac1m$ instead of $\frac1{m-1}$ for the empirical covariance) is answered on Proposition 2.6, page 10 of https://arxiv.org/pdf/2110.06357.pdf .

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Uzu Lim
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This is answered on Proposition 2.6, page 10 of https://arxiv.org/pdf/2110.06357.pdf .