On the fourth moment theorem for the complex multiple Wiener-It\^{o} integrals

In this paper, a product formula of Hermite polynomials is given and then the relation between the real Wiener-It^{o} chaos and the complex Wiener-It^{o} chaos (or: multiple integrals) is shown. By this relation and the known multivariate extension of the fourth moment theorem for the real multiple integrals, the fourth moment theorem (or say: the Nualart-Peccati criterion) for the complex Wiener-It^{o} multiple integrals is obtained.