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

Let $X=\{X_n:n\in \mathbb{N}\}$ be a linear process in which the coefficients are of the form $a_i=i^{-1}\mathrm{\ell }(i)$ with $\mathrm{\ell }$ 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 $$\{\sum _{n=1}^{[Nt]}{X}_n:t\ge 0\}$$ as $N$ tends to infinity, where $[t]$ is the integer part of the non-negative number $t$.