Issue |
EPL
Volume 99, Number 2, July 2012
|
|
---|---|---|
Article Number | 20008 | |
Number of page(s) | 6 | |
Section | General | |
DOI | https://doi.org/10.1209/0295-5075/99/20008 | |
Published online | 24 July 2012 |
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
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