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April 2015 On the Marčenko–Pastur law for linear time series
Haoyang Liu, Alexander Aue, Debashis Paul
Ann. Statist. 43(2): 675-712 (April 2015). DOI: 10.1214/14-AOS1294


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.


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Haoyang Liu. Alexander Aue. Debashis Paul. "On the Marčenko–Pastur law for linear time series." Ann. Statist. 43 (2) 675 - 712, April 2015.


Published: April 2015
First available in Project Euclid: 3 March 2015

zbMATH: 1312.62080
MathSciNet: MR3319140
Digital Object Identifier: 10.1214/14-AOS1294

Primary: 62H25
Secondary: 62M10

Keywords: Autocovariance matrices , Empirical spectral distribution , High-dimensional statistics , linear time series , Marčenko–Pastur law , Stieltjes transform

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 2 • April 2015
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