Regular multidimensional stationary time series

Tamás Szabados

doi:10.1111/jtsa.12611

Abstract

The aim of this article is to give a simpler, more usable sufficient and necessary condition to the regularity of generic weakly stationary time series. Also, this condition is used to show how a regular process can be approximated by a lower rank regular process. The relevance of these issues is shown by the ever increasing presence of high‐dimensional data in many fields lately, and because of this, low rank processes and low rank approximations are becoming more important. Moreover, regular processes are the ones which are completely influenced by random innovations, so they are primary targets both in the theory and applications.

Highlights

Let Xt = (Xt1, . . . , Xtd), t ∈ Z, be a d-dimensional weakly stationary time series, where each Xtj is a complex valued random variable on the same probability space (Ω, F, P)
On it is assumed that μ = 0
The process {Xt} is called regular, if H−∞ = {0} and it is called singular if H−∞ = H(X) := span {Xt : t ∈ Z}

Summary

Introduction

Xtd), t ∈ Z, be a d-dimensional weakly stationary time series, where each Xtj is a complex valued random variable on the same probability space (Ω, F , P). It is classical that any weakly stationary process has a non-negative definite spectral measure matrix dF on [−π, π] such that π. A d-dimensional stationary time series {Xt} is a regular full rank process if and only if (1) it has an absolutely continuous spectral measure matrix dF with density matrix f ; (2) and log det f ∈ L1. The aim of this paper is to give a simpler, more usable sufficient and necessary condition to the regularity of generic time series. This condition is used to show how a regular process can be approximated by a lower rank regular process

Generic regular processes

Low rank approximation