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.