Fourth Moment Theorems for complex Gaussian approximation

We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can be expressed in terms of certain fourth moments of the vector, yielding a quantitative Fourth Moment Theorem for complex Gaussian approximation on complex Markov diffusion chaos. This extends results of Azmoodeh, Campese, Poly (2014) and Campese, Nourdin, Peccati (2015) for the real case. Our main ingredients are a complex version of the so called $\Gamma$-calculus and Stein's method for the multivariate complex Gaussian distribution.