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Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants

Abstract : Let {F_n} be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E[F_n^4] --> E[N^4]=3, where N is a standard Gaussian random variable. Our main result is the following general bound: there exist two finite constants c,C>0 such that, for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < C max(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and h runs over the class of all real functions with a second derivative bounded by 1. This shows that the deterministic sequence max(|E[F_n^3]|, E[F_n^4]-3) completely characterizes the rate of convergence (with respect to smooth distances) in CLTs involving chaotic random variables. These results are used to determine optimal rates of convergence in the Breuer-Major central limit theorem, with specific emphasis on fractional Gaussian noise.
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https://hal.archives-ouvertes.fr/hal-00620384
Contributor : Ivan Nourdin <>
Submitted on : Wednesday, September 7, 2011 - 4:16:39 PM
Last modification on : Friday, April 10, 2020 - 5:24:22 PM
Document(s) archivé(s) le : Thursday, December 8, 2011 - 2:30:22 AM

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  • HAL Id : hal-00620384, version 1
  • ARXIV : 1109.1546

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Hermine Biermé, Aline Bonami, Ivan Nourdin, Giovanni Peccati. Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants. ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2012. ⟨hal-00620384⟩

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