Wishart matrices and Marchenko–Pastur law

Let $X$ have $p$ rows and $n$ columns, with independent standard real Gaussian entries. The columns represent independent mean-zero observations in feature dimension $p$. Define the feature covariance $S=XX^\top/n$ and normalize its empirical measure by $p$. This is a real Wishart model; independent circular complex Gaussian entries with expected absolute square one give its complex counterpart. The division is by sample size, not feature dimension, and the mean is known rather than estimated from the columns.

Let $p,n$ tend to infinity with $p/n\to\gamma\in(0,\infty)$. The empirical measure converges almost surely weakly to the displayed Marchenko–Pastur law. Gaussianity is a sufficient assumption used here, while more general independent-entry theorems require their own moment or truncation conditions. Positivity follows already from the finite identity $u^\top Su=\|X^\top u\|_2^2/n$. No spectral limit is needed to establish that all finite covariance eigenvalues are nonnegative.

When features outnumber samples, the feature covariance has at least $p-n$ zero eigenvalues. Gaussian data have maximal possible rank almost surely, so this zero count is exact. Dividing by feature dimension produces limiting mass $1-1/\gamma$. The continuous part consequently has mass $1/\gamma$ in that regime. When the aspect ratio is below or equal to one, there is no limiting zero atom in this feature-normalized measure, though the density reaches the origin at the square case.

At aspect ratio one, the lower edge is zero and the density has an integrable inverse-square-root singularity there; that is not an atom. More generally the limiting first moment is one under unit entry variance, even when most eigenvalues are zero. A small continuous fraction must then carry a correspondingly larger average nonzero eigenvalue. This separates total spectral mass from conditional mass among nonzero eigenvalues, a distinction that matters for rank estimation, inverse covariance calculations, and comparisons of differently shaped data matrices.

The companion matrix is $X^\top X/n$, now of dimension $n$. Its positive eigenvalues agree with those of the feature matrix, including multiplicity, because both are squared singular values of the same scaled rectangular matrix. Their empirical measures nevertheless differ: one divides the shared eigenvalue list by feature dimension and the other by sample dimension, and each adds its own number of zeros. The displayed transform identity is exact at finite dimensions and remains valid in the limit.

If the aspect ratio is below one, the companion has zero mass $1-\gamma$, even though the feature matrix has no zero atom. For aspect ratio above one, the signed atom term in the measure relation cancels the feature matrix’s excess zeros and the companion has no zeros in the Gaussian full-rank case. The relation should therefore be interpreted as an algebraic identity between complete measures, not as a convex mixture with nonnegative weights for every aspect ratio. Confusing the two normalizations can move an apparent zero atom to the wrong spectral law.