Timeline for Spectrum of sum of weighted Wishart matrices
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 27, 2020 at 20:26 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
link to MSE post
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| Nov 27, 2020 at 17:30 | comment | added | Mikael de la Salle | The idea of symmetrization is that, for independant mean $0$ matrices $Y_i$, up to a factor $2$, the $L^p(\Omega,\mu,C_p^n)$-norm of $\sum_i Y_i$ is of the order the norm of $\sum_i (Y_i - Y_i')$, where $Y'_i$ are independant copies of $Y_i$. By independance, this last quantity as the same norm as $\sum_i \varepsilon_i (Y_i-Y'_i)$ for independant Rademacher $\varepsilon_i$. This is how rademacher functions come into the picture and how NC Khintchine can be used. | |
| Nov 27, 2020 at 17:26 | comment | added | Mikael de la Salle | I do not have the paper under my eyes, but I guess that $L^p(Q,\mu,C_p^n)$ is the space of size-n matrix-valued random variables on a probability space $(\Omega,\mu)$, for the norm $\|X\|^p = \mathbf{E} Tr( |X|^p)$. | |
| Nov 27, 2020 at 17:17 | comment | added | jamblejoe | Thanks @MikaeldelaSalle! Sorry for the confusion. The $p$ in $l_p$ is of course independent of the degree of freedom $p$. In Rudelson's paper, p64/65 where he states the Khintchine inequality, to what sequence are the Rademacher functions applied and what is $\|\cdot\|_{L^p(Q,\mu,C^n_p)}$ space? | |
| Nov 27, 2020 at 15:42 | comment | added | Mikael de la Salle | I am not completely sure I understand your parameters (there are many $p$'s), but I guess that, as in Rudelson's "Random vectors in the isotropic position." (JFA 1999)", you can obtain powerful estimates for the Schatten $p$-norms (=$\ell_p$ norm of the spectrum) by symetrization and using the non-commutative Khintchine inequalities. | |
| Nov 27, 2020 at 12:21 | answer | added | Carlo Beenakker | timeline score: 3 | |
| Nov 27, 2020 at 11:55 | review | First posts | |||
| Nov 27, 2020 at 12:11 | |||||
| Nov 27, 2020 at 11:54 | history | asked | jamblejoe | CC BY-SA 4.0 |