# Spectrum of sum of weighted Wishart matrices

This is a repost from mathstackexchange, as I think asking this question is more appropriate here.

Coming from statistical physics, I am interested in the (real) spectrum of the following sum, and ultimately $$l_p$$ norms of the spectrum, $$$$\sum_{n=1}^m c_n X_n,$$$$ where $$c_n$$ are non-random real numbers and $$X_n$$ are independent Wishart matrices of size $$m\times m$$. Independent in the sense of elementwise independence. The weights $$c_n$$ are nice in the sense that the above sum is uniformly bounded with respect to $$n$$. Let the degree of freedom of the Wishart matrices be $$p$$. I am interested in the limit large $$m$$ and the cases where $$p/m\to 0$$ and $$p/m\to$$ constant > 0. I know that in the latter case, the spectrum of $$X_n$$ will be approximated by the Marchenko-Pastur distribution.

I am new to operator valued probability theory. I did some searching and could find results on spectra of A+B, where A and B are random (free) $$m\times m$$ matrices. Most of them where in the limit of large dimension $$m$$. I could not find something related to summing over orders of the dimension of the matrices $$m$$. Are there any results in that direction, maybe some kind of central limit theorem? If yes, where can I find them?

I am grateful for any advice and literature hints!

• 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 '20 at 15:42
• 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 '20 at 17:17
• 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 '20 at 17:26
• 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 '20 at 17:30

• Thanks! I guess one can still make some use of that result by splitting the $c_j$'s into positive and negative and reduce the problem to the difference of just two wishart matrices. Nov 27 '20 at 21:42