Optimal Gamma Approximation on Wiener Space

Ehsan Azmoodeh; Peter Eichelsbacher; Lukas Knichel

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arXiv · 1902.02658

Optimal Gamma Approximation on Wiener Space

Ehsan Azmoodeh·Peter Eichelsbacher·Lukas Knichel

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Abstract

In \cite{n-p-noncentral}, Nourdin and Peccati 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 $d_2$-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in \cite{n-p-optimal}. 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 D\"obler and Peccati (\cite{d-p}) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound.

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BibTeX RIS

Ehsan Azmoodeh, Peter Eichelsbacher, Lukas Knichel. 2019-02-06. Optimal Gamma Approximation on Wiener Space. https://arxiv.org/abs/1902.02658

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