A Berry-Esse\'en theorem for partial sums of functionals of heavy-tailed moving averages
| Autor: | Mark Podolskij, Andreas Basse-O'Connor, Christoph Thäle |
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| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Pure mathematics Central limit theorem Malliavin calculus Poincaré inequality Stein’s method Poisson distribution symbols.namesake Mathematics::Probability 60E07 Moving average 60F05 linear fractional stable noise 60E07 60F05 60G52 60G57 infinitely divisible processes Mathematics Stein's method Berry–Esseen theorem normal approximation Rate of convergence symbols 60G57 Mathematics [G03] [Physical chemical mathematical & earth Sciences] moving averages Poisson random measures Mathématiques [G03] [Physique chimie mathématiques & sciences de la terre] Statistics Probability and Uncertainty Mathematics - Probability 60G52 |
| Zdroj: | Basse-O'Connor, A, Podolskij, M & Thäle, C 2020, ' A Berry-Esseén theorem for partial sums of functionals of heavy-tailed moving averages ', Electronic Journal of Probability, vol. 25, 31, pp. 1-31 . https://doi.org/10.1214/20-EJP435 Electron. J. Probab. |
| DOI: | 10.1214/20-EJP435 |
| Popis: | 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)]. |
| Databáze: | OpenAIRE |
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