A limit theorem for some linear processes with innovations in the domain of attraction of a stable law

Abstract

Let $X=\{X_n:n\in \mathbb{N}\}$ be a linear process in which the coefficients are of the form $a_i=i^{-1}\ell (i)$ with $\ell$ being a slowly varying function at the infinity and the innovations are independent and identically distributed random variables belonging to the domain of attraction of an $\alpha$-stable law with $\alpha \in (1,2]$. We will establish the asymptotic behavior of the partial sum process \[ \bigg\{\sum\limits_{n=1}^{[Nt]} X_n: t\geq 0\bigg\} \] as $N$ tends to infinity, where $[t]$ is the integer part of the non-negative number $t$.