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

TL;DR

This paper investigates the asymptotic behavior of partial sums of long memory linear processes with infinite variance, revealing limit theorems under certain conditions for functions of such processes.

Contribution

It establishes new limit theorems for functionals of long memory linear processes with infinite variance, extending understanding of their asymptotic properties.

Findings

Derived asymptotic distributions for partial sums

Extended limit theorems to processes with infinite variance

Provided conditions under which the theorems hold

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 $t\geq 0$.