Nonstandard limit theorem for infinite variance functionals

We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long-range dependence is strong enough, the limit is a Hermite process, while for weaker long-range dependence, the limit is alpha-stable Levy motion. For the critical value of the long-range dependence parameter, the limit is a sum of a Hermite process and alpha-stable Levy motion.