Sample covariance matrices and special symmetric partitions

Suppose $X_p$ is a real $p \times n$ matrix with independent entries and consider the (unscaled) sample covariance matrix $S_p=X_pX_p^T$. The Marchenko-Pastur law was discovered as the limit of the bulk distribution of the sample covariance matrix in 1967. There have been extensions of this result in several directions. In this paper, we consider an extension that handles several of the existing ones as well as generates new results. We show that under suitable assumptions on the entries of $X_p$, the limiting spectral distribution exists in probability or almost surely. The moments are described by a set of partitions that are beyond pair partitions and non-crossing partitions and are also related to special symmetric partitions, which are known to appear in the limiting spectral distribution of Wigner-type matrices. Similar results hold for other patterned matrices such as reverse circulant, circulant, Toeplitz, and Hankel matrices.