# A note on the Moment of Complex Wiener-Ito Integrals

## TL;DR

This paper establishes an equivalence in convergence criteria for complex Wiener-Ito integrals using a novel complex Malliavin calculus approach with Wirtinger derivatives.

## Contribution

It introduces a new method of complex Malliavin calculus to directly prove the equivalence between symmetrized and non-symmetrized contraction norm convergence.

## Key findings

- Proves the equivalence of contraction norm convergence in complex Wiener-Ito integrals.
- Develops a new complex Malliavin calculus framework using Wirtinger derivatives.
- Provides a direct proof method for convergence criteria in complex stochastic analysis.

## Abstract

For a sequence of complex Wiener-Ito multiple integrals, the equivalence between the convergence of the symmetrized contraction norms and that of the non-symmetrized contraction norms is shown directly by means of a new version of complex Mallivian calculus using the Wirtinger derivatives of complex-valued functions.