Convergence in variation of the joint laws of multiple Wiener-Itô integrals

The convergence in variation of the laws of multiple Wiener-Itô integrals with respect to their kernel has been studied by Davydov and Martynova in [1987. Limit behavior of multiple stochastic integral. Statistics and Control of Random Process (Preila, 1987), Nauka, Moscow, pp. 55-57 (in Russian)]. Here, we generalize this convergence for the joint laws of multiple Wiener-Itô integrals. In this case, the argument relies on superstructure method which consists in studying related functionals along admissible directions for a Gaussian process.