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Basse-O’Connor, Andreas; Podolskij, Mark; Thäle, Christoph

A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages. (English) Zbl 1456.60051

Electron. J. Probab. 25, Paper No. 31, 31 p. (2020).
Summary: 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 G. Last et al. [Probab. Theory Relat. Fields 165, No. 3–4, 667–723 (2016; Zbl 1347.60012)].

Cited in 5 Documents

MSC:

60E15 Inequalities; stochastic orderings
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
60G52 Stable stochastic processes
60G57 Random measures

Keywords:

central limit theorem; linear fractional stable noise; infinitely divisible processes; Malliavin calculus; moving averages; normal approximation; Poisson random measures; Stein’s method

Citations:

Zbl 1347.60012
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Full Text: DOI arXiv Euclid

References:

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