## The Annals of Statistics

### On the Marčenko–Pastur law for linear time series

#### 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

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

Digital Object Identifier
doi:10.1214/14-AOS1294

Mathematical Reviews number (MathSciNet)
MR3319140

Zentralblatt MATH identifier
1312.62080

#### 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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