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Asymptotic spectral properties and extreme eigenvalues of sample correlation matrices from nonstationary high-dimensional VAR processes

  • Ruihan Liu
  • Chen Wang
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Abstract

We study the limiting spectral distribution and extreme eigenvalues of the sample correlation matrix R̂ from a high-dimensional vector autoregressive (VAR) processes. Depending on whether roots of the characteristic polynomial of given VAR lie on (nonstationary) or outside (super-nonstationary) the unit circle, R̂ presents different types of asymptotic spectral properties. Under both nonstationary and super-nonstationary regimes, the empirical spectral distribution of R̂ degenerates in probability to a point mass at zero. The rescaled extreme eigenvalues of R̂ are asymptotically equivalent to the eigenvalues of a finite-dimensional symmetric random matrix determined by the characteristic roots. Moreover, under super-nonstationarity, R̂ degenerates in probability to a matrix with finite rank.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026asymptotic,
  title = {{Asymptotic spectral properties and extreme eigenvalues of sample correlation matrices from nonstationary high-dimensional VAR processes}},
  author = {Ruihan Liu and Chen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32284},
  eprint = {2610.32284},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32284},
}
RIS
TY  - UNPB
TI  - Asymptotic spectral properties and extreme eigenvalues of sample correlation matrices from nonstationary high-dimensional VAR processes
AU  - Ruihan Liu
AU  - Chen Wang
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32284
UR  - https://arxiv.org/abs/2610.32284
AB  - We study the limiting spectral distribution and extreme eigenvalues of the sample correlation matrix R̂ from a high-dimensional vector autoregressive (VAR) processes. Depending on whether roots of the characteristic polynomial of given VAR lie on (nonstationary) or outside (super-nonstationary) the unit circle, R̂ presents different types of asymptotic spectral properties. Under both nonstationary and super-nonstationary regimes, the empirical spectral distribution of R̂ degenerates in probability to a point mass at zero. The rescaled extreme eigenvalues of R̂ are asymptotically equivalent to the eigenvalues of a finite-dimensional symmetric random matrix determined by the characteristic roots. Moreover, under super-nonstationarity, R̂ degenerates in probability to a matrix with finite rank.
ER  -