Stein's method, Malliavin calculus, Dirichlet forms and the fourth moment theorem

The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and Peccati (2009) using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly (2014) 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 proving the fourth moment theorem.