The Annals of Statistics

On the Marčenko–Pastur law for linear time series

Haoyang Liu, Alexander Aue, and Debashis Paul

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This paper is concerned with extensions of the classical Marčenko–Pastur law to time series. Specifically, $p$-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed (real- or complex-valued) entries possessing zero mean, unit variance and finite fourth moments. The coefficient matrices of the linear process are assumed to be simultaneously diagonalizable. In this setting, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting $p/n\to c\in(0,\infty)$ for which dimension $p$ and sample size $n$ diverge to infinity at the same rate. The results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case.

Article information

Ann. Statist. Volume 43, Number 2 (2015), 675-712.

First available in Project Euclid: 3 March 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Autocovariance matrices empirical spectral distribution high-dimensional statistics linear time series Marčenko–Pastur law Stieltjes transform


Liu, Haoyang; Aue, Alexander; Paul, Debashis. On the Marčenko–Pastur law for linear time series. Ann. Statist. 43 (2015), no. 2, 675--712. doi:10.1214/14-AOS1294.

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Supplemental materials

  • Supplementary material: Supplement to “On the Marčenko–Pastur law for linear time series”. The supplementary material provides additional technical lemmas and their proofs.