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, \begin{equation} \sum_{n=1}^m c_n X_n, \end{equation} 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!