• Long-range dependence;
  • Gaussian process;
  • central limit theorems;
  • non-central limit theorems;
  • asymptotic independence;
  • multiple Wiener–Itô integrals

We study the limit law of a vector made up of normalized sums of functions of long-range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multi-variate Gaussian process involving dependent Brownian motion marginals, (b) a multi-variate process involving dependent Hermite processes as marginals or (c) a combination. We treat cases (a) and (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary, although the conjecture can be resolved in some special cases.