  In this paper we obtain Berry-Esse\'en bounds on partial sums of functionals
of heavy-tailed moving averages, including the linear fractional stable noise,
stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our
rates are obtained for the Wasserstein and Kolmogorov distances, and depend
strongly on the interplay between the memory of the process, which is
controlled by a parameter $\alpha$, and its tail-index, which is controlled by
a parameter $\beta$. In fact, we obtain the classical $\sqrt{1/ n}$ rate of
convergence when the tails are not too heavy and the memory is not too strong,
more precisely, when $\alpha\beta > 3$ or $\alpha\beta > 4$ in the case of
Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely
on a new second-order Poincare inequality on the Poisson space, which we derive
through a combination of Stein's method and Malliavin calculus. This inequality
improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory
Relat. Fields 165 (2016)].
