Frontiers in Mathematical Sciences

The Malliavin-Stein Approach in Probabilistic Approximations

Ehsan Azmoodeh
University of Helsinki


In 2005, Nualart-Peccati discovered a significant result in the realm of probabilistic limit theorems on the Wiener space known as the fourth moment theorem stating that for a sequence $F_n$ of random variables living in a fixed Wiener chaos with variance one, $$F_n \stackrel{\text{law}}{\longrightarrow} N \sim \mathscr{N}(0,1) \quad \text{ if and only if} \quad E(F^4_n) \to 3 \, (= E(N^4)).$$ Recently, these sort of mathematical statements have been extensively got attention and it is culminating in the so-called Malliavin-Stein approach. The talk provides an introduction on the aforementioned approach with emphasis on the breakthrough technique of the Markov generators, spectral properties, and the Gamma calculus introduced by Michael Ledoux.