Optimal Gamma Approximation on Wiener Space.

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. [ABSTRACT FROM AUTHOR]