Spectra of Empirical Auto-Covariance Matrices

Abstract

We compute spectra of sample auto-covariance matrices of second order stationary stochastic processes. We look at a limit in which both the matrix dimension $N$ and the sample size $M$ used to define empirical averages diverge, with their ratio $\alpha =N/M$ kept fixed. We find a remarkable scaling relation which expresses the spectral density $\rho (\lambda )$ of sample auto-covariance matrices for processes with dynamical correlations as a continuous superposition of appropriately rescaled copies of the spectral density ${\rho }_{\alpha }^{(0)}(\lambda )$ for a sequence of uncorrelated random variables. The rescaling factors are given by the Fourier transform $\hat{C}(q)$ of the auto-covariance function of the stochastic process. We also obtain a closed-form approximation for the scaling function ${\rho }_{\alpha }^{(0)}(\lambda )$. This depends on the shape parameter $\alpha$, but is otherwise universal: it is independent of the details of the underlying random variables, provided only they have finite variance. Our results are corroborated by numerical simulations using auto-regressive processes.

4 pages, 2 figures

References in corpus (3)

• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble
• On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices

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