Limit theorems for functionals of long memory linear processes with infinite variance

Abstract

Let $X=\{X_n:n\in \mathbb{N}\}$ be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an $\alpha$-stable law with $\alpha \in (0,2)$. Then, for any integrable and square integrable function $K$ on $\mathbb{R}$, under certain mild conditions, we establish the asymptotic behavior of the partial sum process \[ \left\{\sum\limits_{n=1}^{[Nt]}\big[K(X_n)-\E K(X_n)\big]:\; t\geq 0\right\} \] as $N$ tends to infinity, where $[Nt]$ is the integer part of $Nt$ for $t\ge 0$.