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2021 Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series
Philippe Loubaton, Alexis Rosuel
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Electron. J. Statist. 15(2): 5380-5454 (2021). DOI: 10.1214/21-EJS1923

Abstract

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series (yn)nZ with independent components is studied under the asymptotic regime where the sample size N converges towards + while the dimension M of y and the smoothing span of the estimator grow to infinity at the same rate in such a way that MN0. It is established that, at each frequency, the estimated spectral coherency matrix is close to the sample covariance matrix of an independent identically NC(0,IM) distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of 1M+MN+(MN)3 where N is the sample size. Numerical simulations supports our results.

Funding Statement

This work is funded by ANR Project HIDITSA, reference ANR-17-CE40-0003.

Citation

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Philippe Loubaton. Alexis Rosuel. "Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series." Electron. J. Statist. 15 (2) 5380 - 5454, 2021. https://doi.org/10.1214/21-EJS1923

Information

Received: 1 July 2020; Published: 2021
First available in Project Euclid: 15 December 2021

Digital Object Identifier: 10.1214/21-EJS1923

Subjects:
Primary: 60B20 , 62H15
Secondary: 62M15

Keywords: high dimensional statistics , Independence test , random matrices , spectral analysis , time series

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Vol.15 • No. 2 • 2021
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