  Recently a functional limit theorem for sums of moving averages with random
coefficients and i.i.d. heavy tailed innovations has been obtained under the
assumption that all partial sums of the series of coefficients are a.s. bounded
between zero and the sum of the series. The convergence takes place in the
space $D[0,1]$ of c\`{a}dl\`{a}g functions with the Skorohod $M_{2}$ topology.
In this article we extend this result to the case when the innovations are
weakly dependent in the sense of strong mixing and local dependence condition
$D'$.
