Multivariate limits of multilinear polynomial-form processes with long memory

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.