## Electronic Journal of Statistics

### Spectral analysis of high-dimensional time series

#### Abstract

A useful approach for analysing multiple time series is via characterising their spectral density matrix as the frequency domain analog of the covariance matrix. When the dimension of the time series is large compared to their length, regularisation based methods can overcome the curse of dimensionality, but the existing ones lack theoretical justification. This paper develops the first non-asymptotic result for characterising the difference between the sample and population versions of the spectral density matrix, allowing one to justify a range of high-dimensional models for analysing time series. As a concrete example, we apply this result to establish the convergence of the smoothed periodogram estimators and sparse estimators of the inverse of spectral density matrices, namely precision matrices. These results, novel in the frequency domain time series analysis, are corroborated by simulations and an analysis of the Google Flu Trends data.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 4079-4101.

Dates
First available in Project Euclid: 9 October 2019

https://projecteuclid.org/euclid.ejs/1570586483

Digital Object Identifier
doi:10.1214/19-EJS1621

Mathematical Reviews number (MathSciNet)
MR4017528

Zentralblatt MATH identifier
07116197

#### Citation

Fiecas, Mark; Leng, Chenlei; Liu, Weidong; Yu, Yi. Spectral analysis of high-dimensional time series. Electron. J. Statist. 13 (2019), no. 2, 4079--4101. doi:10.1214/19-EJS1621. https://projecteuclid.org/euclid.ejs/1570586483

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