# Exchangeable pairs on Wiener chaos

## TL;DR

This paper explores the connection between Malliavin calculus and Stein's method through exchangeable pairs of Brownian motions, providing new proofs and extending results to multidimensional cases in the context of Wiener chaos.

## Contribution

It constructs exchangeable pairs of Brownian motions linking Malliavin operators with exchangeable pairs, offering a new proof of the fourth moment theorem and extending it to multiple dimensions.

## Key findings

- New construction of exchangeable pairs of Brownian motions.
- Alternative proof of the quantitative fourth moment theorem.
- Extension of results to multidimensional Wiener chaos.

## Abstract

In [14], Nourdin and Peccati combined the Malliavin calculus and Stein's method of normal approximation to associate a rate of convergence to the celebrated fourth moment theorem [19] of Nualart and Peccati. Their analysis, known as the Malliavin-Stein method nowadays, has found many applications towards stochastic geometry, statistical physics and zeros of random polynomials, to name a few. In this article, we further explore the relation between these two fields of mathematics. In particular, we construct exchangeable pairs of Brownian motions and we discover a natural link between Malliavin operators and these exchangeable pairs. By combining our findings with E. Meckes' infinitesimal version of exchangeable pairs, we can give another proof of the quantitative fourth moment theorem. Finally, we extend our result to the multidimensional case.