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In this paper we obtain Berry-Esse\'en bounds on partial sums of functionalsof heavy-tailed moving averages, including the linear fractional stable noise,stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Ourrates are obtained for the Wasserstein and Kolmogorov distances, and dependstrongly on the interplay between the memory of the process, which iscontrolled by a parameter $\alpha$, and its tail-index, which is controlled bya parameter $\beta$. In fact, we obtain the classical $\sqrt{1/ n}$ rate ofconvergence 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 ofWasserstein and Kolmogorov distance, respectively. Our quantitative bounds relyon a new second-order Poincare inequality on the Poisson space, which we derivethrough a combination of Stein's method and Malliavin calculus. This inequalityimproves and generalizes a result by Last, Peccati, Schulte [Probab. TheoryRelat. Fields 165 (2016)].

A Berry-Esse\'en theorem for partial sums of functionals of heavy-tailed moving averages