# Weak convergence on Wiener space: targeting the first two chaoses

## TL;DR

This paper establishes necessary and sufficient conditions for the convergence in law of sequences of random variables in the sum of the first two Wiener chaoses, using Malliavin calculus and Gamma-operators, extending prior results.

## Contribution

It provides a complete characterization of convergence in law for sequences in the first two Wiener chaoses, including multiple Wiener integrals, and explores stable convergence and limitations on target variables.

## Key findings

- Conditions for convergence in law are characterized precisely.
- Results extend previous work by Azmoodeh, Peccati, and Poly (2014).
- Certain classes of target variables are shown to be unattainable.

## Abstract

We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Gamma-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence situations. Our methods are applied as well to investigate stable convergence. We finally exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos.