Multivariate limits of multilinear polynomial-form processes with long memory

We consider the multilinear polynomial-form process $X(n)=\sum_{1\le i_1<\ldots<i_k<\infty}a_{i_1}\ldots a_{i_k}ε_{n-i_1}\ldotsε_{n-i_k},$ obtained by applying a multilinear polynomial-form filter to i.i.d.\ sequence $\{ε_i\}$ 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 $\{ε_i\}$. 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.

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