Optimal Gamma Approximation on Wiener Space



Azmoodeh, Ehsan ORCID: 0000-0002-0401-793X, Eichelsbacher, Peter and Knichel, Lukas
(2020) Optimal Gamma Approximation on Wiener Space. Alea (Rio de Janeiro): Latin American journal of probability and mathematical statistics, 17 (1). pp. 101-132.

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Abstract

Nourdin and Peccati (2009a) established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in Nourdin and Peccati (2015). In order to achieve our goal, we introduce a novel operator theory approach to Stein’s method. The recent development in Stein’s method for the Gamma distribution of Döbler and Peccati (2018) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound.

Item Type: Article
Additional Information: arXiv admin note: text overlap with arXiv:1806.03878
Uncontrolled Keywords: gamma approximation, Weiner chaos, cumulants/moments, weak convergence, Malliavin calculus, Berry–Esseen type bounds, Stein’s method, Smooth Wasserstein distances, quadratic form
Depositing User: Symplectic Admin
Date Deposited: 05 May 2020 10:33
Last Modified: 18 Jan 2023 23:58
DOI: 10.30757/ALEA.v17-05
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3078471