A Note on Mixing in High Dimensional Time Series

TL;DR

This paper investigates mixing conditions in high-dimensional time series, establishing methods to verify $ ho$-mixing and demonstrating that VAR(1) and VARMA(1,1) models satisfy this condition under low-rank assumptions.

Contribution

It provides a novel approach to verify $ ho$-mixing in high-dimensional time series using Pearson's ${phi}^2$, and proves VAR and VARMA models meet this condition in low-rank settings.

Findings

$ ho$-mixing can be verified using Pearson's ${phi}^2$ in high dimensions.

VAR(1) and VARMA(1,1) models satisfy $ ho$-mixing under low-rank assumptions.

The paper links mixing conditions with practical high-dimensional time series models.

Abstract

Various mixing conditions have been imposed on high dimensional time series, including the strong mixing ($\alpha$-mixing), maximal correlation coefficient ($\rho$-mixing), absolute regularity ($\beta$-mixing), and $\phi$-mixing. $\alpha$-mixing condition is a routine assumption when studying autoregression models. $\rho$-mixing can lead to $\alpha$-mixing. In this paper, we prove a way to verify $\rho$-mixing under a high-dimensional triangular array time series setting by using the Pearson's $\phi^2$, mean square contingency. Vector autoregression model VAR(1) and vector autoregression moving average VARMA(1,1) are proved satisfying $\rho$-mixing condition based on low rank setting.