Multivariate limits of multilinear polynomial-form processes with long memory

TL;DR

This paper investigates the asymptotic behavior of multivariate polynomial-form processes with long memory, revealing that their normalized partial sums converge to either Gaussian, Hermite, or mixed processes, depending on the filter parameters.

Contribution

It characterizes the multivariate limits of polynomial-form processes with long memory, identifying conditions for Gaussian, Hermite, or mixed limit processes, and determines the independence structure.

Findings

Limit processes are Gaussian, Hermite, or mixtures.

The convergence depends on the filter's memory properties.

Independent components of the limit vectors are explicitly identified.

Abstract

We consider the multilinear polynomial-form process \[X(n)=\sum_{1\le i_1<\ldots<i_k<\infty}a_{i_1}\ldots a_{i_k}\epsilon_{n-i_1}\ldots\epsilon_{n-i_k},\] obtained by applying a multilinear polynomial-form filter to i.i.d.\ sequence $\{\epsilon_i\}$ where $\{a_i\}$ is regularly varying. The resulting sequence $\{X(n)\}$ will then display either short or long memory. Now consider a vector of such X(n), whose components are defined through different $\{a_i\}$'s, that is, through different multilinear polynomial-form filters, but using the same $\{\epsilon_i\}$. What is the limit of the normalized partial sums of the vector? We show that the resulting limit is either a) a multivariate Gaussian process with Brownian motion as marginals, or b) a multivariate Hermite process, or c) a mixture of the two. We also identify the independent components of the limit vectors.