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Local Law and Outlier Eigenvalues of Spiked Separable Covariance Matrices

  • Zhili Wang
  • Bin Qin
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Abstract

We prove local laws for the resolvents of separable covariance matrices of the form 𝒬=A1/2XBX*A1/2, where X=(xij) is a p× n random matrix whose entries xij are i.i.d.~random variables with mean 0 and variance n−1, and A,B are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of 𝒬 and use it to prove optimal local laws without the technical assumption 𝔼[xij3]=0, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026local,
  title = {{Local Law and Outlier Eigenvalues of Spiked Separable Covariance Matrices}},
  author = {Zhili Wang and Bin Qin},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29872},
  eprint = {2608.29872},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29872},
}
RIS
TY  - UNPB
TI  - Local Law and Outlier Eigenvalues of Spiked Separable Covariance Matrices
AU  - Zhili Wang
AU  - Bin Qin
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29872
UR  - https://arxiv.org/abs/2608.29872
AB  - We prove local laws for the resolvents of separable covariance matrices of the form 𝒬=A^1/2XBX^*A^1/2, where X=(x_ij) is a p× n random matrix whose entries x_ij are i.i.d.~random variables with mean 0 and variance n^−1, and A,B are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of 𝒬 and use it to prove optimal local laws without the technical assumption 𝔼[x_ij^3]=0, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.
ER  -