### Stein's method, Malliavin calculus, Dirichlet forms and the fourth moment theorem [chapter]

Louis H.Y. Chen, Guillaume Poly
2014 Festschrift Masatoshi Fukushima
The fourth moment theorem provides error bounds in the central limit theorem for elements of Wiener chaos of any order. It was proved by Nourdin and Peccati [31] using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly [3] using Stein's method and Dirichlet forms. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in
more » ... are exploited in proving the fourth moment theorem. 2 The success of such a connection relies on the fact that both Stein's method and the Malliavin calculus are built on some integration by parts techniques. In addition, the operators of the Malliavin calculus, D, δ, L, satisfy several nice integration by parts formulae which fit in perfectly with the so-called Stein equation. For a good overview of these techniques, we refer to the following website. https://sites.google.com/site/malliavinstein/home The work of Nourdin and Peccati [31] has added a new dimension to Stein's method. Their approach of combining Stein's method with the Malliavin calculus has led to improvements and refinements of many results in probability theory, such as the Breuer-Major theorem [9] . More recently, this approach has been successfully used to obtain central limit theorems in stochastic geometry, stochastic calculus, statistical physics, and for zeros of random polynomials. It has also been extended to different settings as in non-commutative probability and Poisson chaos. Of particular interest is the connection between the Nourdin-Peccati analysis and information theory, which was recently revealed in [26, 30] . An overview of these new developments can also be found in the above website. The approach of Nourdin and Peccati [31] entails the use of the so-called product formula for Wiener integrals. The use of this formula makes the proofs rather involved since it relies on subtle combinatorial arguments. Very recently, starting with the work of Ledoux [25], a new approach to the fourth moment theorem was developed by Azmoodeh, Campese and Poly [2] by combining Stein's method with the Dirichlet form calculus. An advantage of this new approach is that it provides a simpler proof of the theorem by avoiding completely the use of the product formula. Moreover, since a Dirichlet space is a more general concept than the Wiener space, the former contains examples of fourth moment theorems that cannot be realized on the latter. A more algebraic flavor of this approach has enabled Azmoodeh, Malicet and Poly [3] to prove that convergence of pairs of moments other than the 2nd and 4th (for example, the 6th and 68th) also implies the central limit theorem. This new approach seems to open up new possibilities and perhaps also central limit theorems on manifolds. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in proving the fourth moment theorem. 2. STEIN'S METHOD 2.1. How it began. Stein's method began with Charles Stein using his own approach in the 1960's to prove the combinatorial central limit theorems of Wald and Wolfowitz [43] and of Hoeffding [23]. Motivated by permutation tests in nonparametric statistics, Wald and Wolfowitz [43] proved that under certain conditions,