On the Rate of Convergence to a Gamma Distribution on Wiener Space

Abstract

In [NP09a], 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 investigate the rate of convergence in Gamma approximation on Wiener chaos in terms of the iterated Gamma operators of Malliavin Calculus. On the second Wiener chaos, our upper bound can be further extended to an exact rate of convergence in a suitable probability metric $d_2$ in terms of the maximum of the third and fourth cumulants, analogous to that of normal approximation in [NP15] under one extra mild condition. We end the paper with some novel Gamma characterization within the second Wiener chaos as well as Gamma approximation in Kolmogorov distance relying on the classical Berry-Esseen type inequality.