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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 $α=N/M$ kept fixed. We find a remarkable scaling relation which expresses the spectral density $ρ(λ)$ of sample auto-covariance matrices for processes with dynamical correlations as a continuous superposition of appropriately rescaled copies of the spectral density $ρ^{(0)}_α(λ)$ 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 $ρ^{(0)}_α(λ)$. This depends on the shape parameter $α$, 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.