2019/02/06 by Azmoodeh, Ehsan, Eichelsbacher, Peter, Knichel, Lukas
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1902.02658
In \citen-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 d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in \citen-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 Döbler and Peccati (\cited-p) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound.