Limiting ESD for a weighted sample covariance matrix

We study the empirical spectral distribution (ESD) of the random matrix $M_N=\frac1N\sum_{i=1}^N f(|πY_i|^2)\,Y_iY_i^\top$, where $π$ is a rank $k$ orthogonal projector on $\R^p$, $Y_1,\dots,Y_N$ are i.i.d. standard Gaussian vectors, and $k\to \infty$, $p/N\to γ$. We here assume that $f(x)=g(x/k)$ for some function $g$ discontinuous at $x=1$. We show that the asymptotic ESD is characterized by a Marchenko--Pastur equation, independently of $k$. We also show that the result extends to sub Gaussian random vectors $Y_i$ and to the case where the projector is replaced with a more general weight $ϕ(Y_i)$. Last, in the Gaussian case, we study the emergence of outliers due to the weighting.