The Annals of Statistics

On the Marčenko–Pastur law for linear time series

Haoyang Liu, Alexander Aue, and Debashis Paul

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Abstract

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

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

Dates
First available in Project Euclid: 3 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1425398505

Digital Object Identifier
doi:10.1214/14-AOS1294

Mathematical Reviews number (MathSciNet)
MR3319140

Zentralblatt MATH identifier
1312.62080

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1425398505.


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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.