Spectra of Empirical Auto-Covariance Matrices

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 $\rho^{(0)}_\alpha(\lambda)$ 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 $\rho^{(0)}_\alpha(\lambda)$. 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.