Reimer Kuehn (KCL)

Spectra of empirical auto-covariance matrices derived from time series

Abstract:
We compute spectral densities of empirical auto-covariance matrices of second order stationary stochastic processes in the limit where the number of time-lags included is large. Matrices of this type constitute one way of randomizing Toeplitz matrices. While not Toeplitz themselves, their averages are, and fluctuations about these averages decrease with increasing sample size M. 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. One of our main results is a remarkable scaling relation which relates spectral densities for processes with correlations via a simple integral to the corresponding spectral densities for processes without correlations (i.e. for sequences of i.i.d variables), and we derive a closed form approximation for the latter. Our results are well corroborated by simulations using auto-regressive processes.