Nonstandard limit theorem for infinite variance functionals

TL;DR

This paper establishes nonstandard limit theorems for functionals of long-range dependent Gaussian sequences with infinite variance, revealing different limiting processes based on the strength of dependence.

Contribution

It introduces new limit theorems for infinite variance functionals under long-range dependence, identifying conditions for Hermite and stable Lévy process limits.

Findings

Limit is a Hermite process under strong dependence

Limit is an α-stable Lévy motion under weaker dependence

Critical dependence yields a sum of Hermite and stable Lévy processes

Abstract

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.