I consider a random matrix of the type : $ M_n = \frac{1}{n} X_n D_n X_n^\intercal \in \mathbb{R}^{n \times n}$ , in which all matrices are square of size $ n$ . $ D_n$ is a deterministic diagonal matrix with elements $ D_{n,\mu}$ which are bounded : $ $ \sup_\mu |D_{n,\mu}| \leq \rho < +\infty.$ $ The matrix elements of the matrix $ X_n$ are standard i.i.d. Gaussian variables (mean zero, variance $ 1$ ). I consider a function $ f : \mathbb{R} \to \mathbb{R}$ which is Lipschitz (I can basically make any other reasonable assumption on $ f$ ), and I’m interested in the concentration, as $ n$ grows, of the linear spectral statistic: $ $ G_{n,f}(X_n) = \frac{1}{n} \mathrm{Tr} f(M_n)$ $

There are some classic RMT results (for instance from the book of Anderson,Guionnet,Zeitouni : Link (Section 2.3 or 4.4), or in the previous paper of Guionnet&Zeitouni Link (these results can also be found in many other places).

For instance (Corollary 1.8.b of the Guionnet-Zeitouni), if one assumes that all the elements of $ D_n$ are positive, and that $ x \mapsto f(x^2)$ is Lipschitz with Lipschitz constant $ L > 0$ , then one has: $ $ \mathbb{P}\left(\left|G_{n,f}(X_n) – \mathbb{E}G_{n,f}(X_n)\right| \geq t\right) \leq \exp\left(- \frac{1}{2 \rho L}n^2 t^2\right) $ $

I have found other similar results in the litterature, *but I couldn’t find anything concerning the case of non-positive $ D_n$ *. My main question is : does there exists similar concentration results concerning this case ? I don’t know if the lack of study concerning this case is due to a real theoretical difficulty or if it comes from the fact that these matrices are less natural as they are not empirical covariance matrices…

Any help would be appreciated !

Thanks a lot ðŸ™‚