New probabilistic approximations for non-linear functionals of random elds and random measures
1) A novel collection of analytical inequalities – generalizing the second order Poincaré estimates developed by Chatterjee (2009) and Nourdin, Peccati and Reinert (2010) – allowing one to prove presumably optimal rates of convergence in limit theorems involving non-linear functionals of Gaussian fields. These findings fill several gaps left open in the 2010 paper quoted above.
2) A definitive version – in any dimension – of the fourth moment theorem for eigenfunctions of the Ornstein-Uhlenbeck generator on the Poisson space. This contribution completes in a substantial and striking way a recent work by Döbler and Peccati, and virtually concludes a circle of ideas (related to central limit results on the Poisson space, via Stein’s method and Malliavin calculus) initiated in 2010, in a paper by Peccati, Solé, Taqqu and Utzet. One of the novel crucial contributions of this work (jointly written with Döbler and Zheng) is the introduction of a local version of the exchangeable pairs approach on configuration spaces, relying on pioneering ideas by E. Mecke.
3) Several results concerning the convergence in distribution (in the infinite-dimensional functional sense) of geometric functionals of Gaussian random waves on $R^2$, with specific emphasis on problems of finite-dimensional distrbution and tightness. Such a collection of findings complements a recent work by Nourdin, Peccati and Reinert, about the CLTs for the nodal lenght of Berry’s random wave mode