  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 L\'{e}vy motion. For the
critical value of the long-range dependence parameter, the limit is a sum of a
Hermite process and $\alpha$-stable L\'{e}vy motion.
