Summary: We consider the partial-sum process \(\sum_{k=1}^{[nt]}X_k^{(n)}\), where \(\{ X_k^{(n)}=\sum_{j=0}^{\infty}\alpha_j^{(n)}\xi_{k-j}(b(n)), k\in\mathbb{Z}\}, n\geqslant 1\), is a series of linear processes with tapered filter \(\alpha_j^{(n)} =\alpha_j \mathbf{1}_{\{0\leqslant j\leqslant\lambda (n)\}}\) and heavy-tailed tapered innovations \(\xi_j (b(n)), j \in \mathbb{Z}\). Both tapering parameters \(b(n)\) and \(\lambda (n)\) grow to \(\infty\) as \(n \rightarrow \infty\). The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter \(a_i, i \geqslant 0\), and nontapered innovations. We consider the cases where \(b(n)\) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of \(\lambda (n)\) (strong, weak, and moderate tapering).