A note on a Mar\v{c}enko-Pastur type theorem for time series

TL;DR

This paper extends the Marčenko-Pastur theorem to time series with temporal correlations, providing an explicit characterization of the spectral distribution of sample covariance matrices and a numerical method for density computation.

Contribution

It introduces a novel extension of the Marčenko-Pastur theorem to correlated time series, with explicit equations and a numerical algorithm for spectral density estimation.

Findings

Derived an explicit equation for the LSD's Stieltjes transform

Provided a numerical algorithm for density computation

Extended the theorem to time series with spectral density dependence

Abstract

In this note we develop an extension of the Mar\v{c}enko-Pastur theorem to time series model with temporal correlations. The limiting spectral distribution (LSD) of the sample covariance matrix is characterised by an explicit equation for its Stieltjes transform depending on the spectral density of the time series. A numerical algorithm is then given to compute the density functions of these LSD's.