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arXiv · 2610.32284

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 $\hat{\boldsymbol{R}}$ 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, $\hat{\boldsymbol{R}}$ presents different types of asymptotic spectral properties. Under both nonstationary and super-nonstationary regimes, the empirical spectral distribution of $\hat{\boldsymbol{R}}$ degenerates in probability to a point mass at zero. The rescaled extreme eigenvalues of $\hat{\boldsymbol{R}}$ are asymptotically equivalent to the eigenvalues of a finite-dimensional symmetric random matrix determined by the characteristic roots. Moreover, under super-nonstationarity, $\hat{\boldsymbol{R}}$ degenerates in probability to a matrix with finite rank.

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Ruihan Liu, Chen Wang. 2026-10-23. Asymptotic spectral properties and extreme eigenvalues of sample correlation matrices from nonstationary high-dimensional VAR processes. https://arxiv.org/abs/2610.32284

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