Eigenvector overlaps of sample covariance matrices with intersecting time periods

Battle Count: 81

Highly relevant for quantitative finance practitioners. The paper provides an exact null-hypothesis benchmark for testing whether observed changes in covariance matrix eigenvectors between overlapping time windows are due to measurement noise or genuine market regime shifts. This is directly applicable to: (1) deciding when to re-estimate covariance matrices for portfolio construction, (2) detecting sudden regime shifts using sliding windows, (3) understanding the Ledoit-Peche eigenvalue cleaning formula, and (4) assessing the reliability of risk models over time. The overlap parameter t naturally arises in rolling-window estimation common in practice.

The theoretical formula (Eq. 28) is explicit and can be implemented numerically by solving the coupled system of equations for m(z) and M(z) (Marcenko-Pastur type equations) and then evaluating the overlap expression. However, the underlying proof framework (Girko linearization, Dyson equations, multi-resolvent local laws, stability operators) is extremely technical and requires advanced knowledge of random matrix theory. For practical use, implementing just the final formula is moderate complexity; understanding and extending the theory is very high complexity.

The paper provides the full analytical formula (Eq. 28) and numerical parameters (N=300, q=1/2, t values, number of repetitions). However, no code repository is mentioned. The Supplementary Material contains detailed derivations. Reproducing the numerical experiments requires implementing the theoretical formula and generating synthetic/real financial data. The bootstrapping procedure for real data is described but not fully specified in code.