On the Marčenko–Pastur law for linear time series
Haoyang Liu, Alexander Aue, Debashis Paul
The Annals of Statistics · 2015 · Volume 43 · Issue 2
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.
Keywords
62H25, 62M10, Autocovariance matrices, Empirical spectral distribution, High-dimensional statistics, linear time series, Marčenko–Pastur law, Stieltjes transform
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