Simulate the Marchenko–Pastur law

CodingIntermediate⏱ 1 hourRandom Matricessimulationmarchenko-pasturuniversalitycovariance

Theory: Sample Covariance and the Marchenko–Pastur Law →

Problem

Generate $X\in\R^{n\times p}$ with i.i.d. unit-variance entries, form $\hat\Sigma = \frac1nX^\top X$ , and compare the eigenvalue histogram with the Marchenko–Pastur density.

Sweep $\lambda = p/n$ and check the edges.
Replace Gaussian entries with Rademacher and with a heavy-tailed unit-variance law; comment on the bulk and on the extreme eigenvalues.
Add a rank-one spike $\Sigma = I + h\,vv^\top$ and find empirically the value of $h$ at which the top eigenvalue separates from the bulk. Compare with $\sqrt\lambda$ .

Implement it, then check the implementation against something you can verify independently.

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Solution

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