  In the paper we consider the partial sum process
$\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}, \ k\in Z\},\ n\ge 1,$ is a
series of linear processes with innovations having heavy-tailed tapered
distributions with tapering parameter $b_n$ depending on $n$. It is shown that,
depending on the properties of a filter of a linear process under consideration
and on the parameter $b_n$ defining if the tapering is hard or soft, the limit
process for such partial sum process can be fractional Brownian motion or
linear fractional stable motion.
