We consider a class of regularly-varying stochastic processes with long-range dependence introduced first by Rosinski and Samorodnitsky in 1996. The original model can be represented as stochastic integrals with respect to a symmetric alpha-stable random measure, and we investigate an extension of the model that can be represented as multiple-stochastic integrals. When the multiplicity p is greater than or equal to 2 a new phase transition occurs. In the sub-critical regime, we reveal a delicate asymptotic behavior of extremes: the so-called candidate extremal index and the extremal index are not the same. We are not aware of any other regularly varying processes with such a feature.

Based on joint works with Shuyang Bai and Rafal Kulik.