  We study convergence in law of partial sums of linear processes with
heavy-tailed innovations. In the case of summable coefficients necessary and
sufficient conditions for the finite dimensional convergence to an
$\alpha$-stable L\'evy Motion are given. The conditions lead to new, tractable
sufficient conditions in the case $\alpha \leq 1$. In the functional setting we
complement the existing results on $M_1$-convergence, obtained for linear
processes with nonnegative coefficients by Avram and Taqqu (1992) and improved
by Louhichi and Rio (2011), by proving that in the general setting partial sums
of linear processes are convergent on the Skorokhod space equipped with the $S$
topology, introduced by Jakubowski (1997).
