Optimal Gamma Approximation on Wiener Space

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.