Spectra of empirical auto-covariance matrices

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Abstract

We compute spectral densities of large sample auto-covariance matrices of stationary stochastic processes at fixed ratio alpha = N/M of matrix dimension N and sample size M. We find a remarkable scaling relation which expresses the spectral density rho(alpha)(lambda) of sample auto-covariance matrices for processes with correlations as a continuous superposition of copies of the spectral density rho((0))(alpha)(lambda) for a sequence of uncorrelated random variables at the same value of alpha, rescaled in terms of the Fourier transform (C) over cap (q) of the true auto-covariance function. We also obtain a closed-form approximation for the scaling function rho((0))(alpha)(lambda). Our results are in excellent agreement with numerical simulations using auto-regressive processes, and processes exhibiting a power-law decay of correlations. Copyright (C) EPLA, 2012

Original languageEnglish
Article number20008
Pages (from-to)-
Number of pages6
JournalEUROPHYSICS LETTERS
Volume99
Issue number2
DOIs
Publication statusPublished - Jul 2012

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