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

TL;DR

This paper extends the fourth moment theorem to complex Wiener-Itô multiple integrals by establishing a product formula for Hermite polynomials and relating real and complex chaos.

Contribution

It introduces a relation between real and complex Wiener-Itô chaos and proves a fourth moment theorem for complex multiple integrals.

Findings

Established a product formula for Hermite polynomials.

Connected real and complex Wiener-Itô chaos.

Proved the fourth moment theorem for complex integrals.

Abstract

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.