  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 $t$.
