  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 multivariate Gaussian process involving
dependent Brownian motion marginals, or (b) a multivariate process involving
dependent Hermite processes as marginals, or (c) a combination. We treat cases
(a), (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.
