Timeline for Density of eigenvalues of empirical covariance matrix of vectors uniform on the sphere
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2023 at 22:07 | answer | added | Carlo Beenakker | timeline score: 1 | |
| Mar 20, 2023 at 21:53 | comment | added | user27182 | Is an asymptotic result available for this setting, if an exact one is not? | |
| Mar 20, 2023 at 21:22 | comment | added | Carlo Beenakker | in your case the martrix elements are correlated, since the squares must sum to $d$, that is an essential complication compared to the case of independent Gaussians; for $d\gg 1$ the correlations can be neglected, but not for small $d$. | |
| Mar 20, 2023 at 21:22 | comment | added | user27182 | Maybe it's possible to get somewhere using theorem 1 of these notes galton.uchicago.edu/~lalley/Courses/386/ClassicalEnsembles.pdf but in my case it seems diagonal entries don't have a density | |
| Mar 20, 2023 at 20:51 | comment | added | user27182 | Yes, the rows are independent. I'm after the exact value for fixed $n$ and $d$. It is known when the rows are gaussian random vectors. E.g. Corollary 3.2.19 of Robb J Muirhead. Aspects of multivariate statistical theory. Vol. 197. John Wiley & Sons, 2009. I would expect it to be known in the uniform case too, right? | |
| Mar 20, 2023 at 20:42 | history | edited | user27182 | CC BY-SA 4.0 |
added 7 characters in body
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| Mar 20, 2023 at 18:03 | comment | added | Carlo Beenakker | are the rows independent? closed-form expressions for arbitrary $n,d$ are unlikely; are you interested in asymptotics for $n,d\gg 1$? | |
| Mar 20, 2023 at 17:26 | history | asked | user27182 | CC BY-SA 4.0 |