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I am reading about the Marchenko-Pastur Law (https://people.math.wisc.edu/~valko/courses/833/2009f/lec_6_7.pdf) and trying to decipher the main theorem so it is more intituitive.

Would it mean in any case that if $p/n \rightarrow 1$ (or $y$ is close to 1), then the singular values of a matrix $X$ (or the eigenvalues of $XX'$) are more likely to include one that is close to 0?

kloop
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  • Yes, you'll have small singular values in a square matrix, approaching 0 as d grows at a rate of something like 1/d. For rectangular matrix, they'll be bounded away from 0 – Yaroslav Bulatov Feb 08 '24 at 18:19

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