A Berry–Esseén theorem for partial sums of functionals of heavy-tailed moving averages

Abstract

In this paper we obtain Berry–Esseén 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 $1/\sqrt {n}$ 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 Poincaré 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)].