Spectra of empirical auto-covariance matrices
Department of Mathematics, King's College London - Strand, London WC2R 2LS, UK
Received: 29 March 2012
Accepted: 26 June 2012
We compute spectral densities of large sample auto-covariance matrices of stationary stochastic processes at fixed ratio α = N/M of matrix dimension N and sample size M. We find a remarkable scaling relation which expresses the spectral density ρα(λ) of sample auto-covariance matrices for processes with correlations as a continuous superposition of copies of the spectral density ρ(0)α(λ) for a sequence of uncorrelated random variables at the same value of α, rescaled in terms of the Fourier transform of the true auto-covariance function. We also obtain a closed-form approximation for the scaling function ρ(0)α(λ). Our results are in excellent agreement with numerical simulations using auto-regressive processes, and processes exhibiting a power-law decay of correlations.
PACS: 02.50.-r – Probability theory, stochastic processes, and statistics / 05.10.-a – Computational methods in statistical physics and nonlinear dynamics
© EPLA, 2012