We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. . In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.Finally, a transfer principle "from-Poisson-to-Gaussian" is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.where d TV denotes the total variation distance between the laws of two real random variables. The techniques developed in [26] have also been adapted to non-Gaussian spaces which admit a Malliavin calculus structure: for instance, the papers [16,34,36,41] deal with the Poisson space case, whereas [17,18,30,44] develop the corresponding techniques for sequences of independent Rademacher random variables. The question FOURTH MOMENT THEOREMS 3 about general fourth moment theorems on these spaces, however, has remained open in general, until the two recent articles [13] and [11].

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We present an improved version of the second order Gaussian Poincaré inequality, firstly introduced in Chatterjee (2009) and Nourdin, Peccati and Reinert (2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative CLTs for non-linear functionals of stationary Gaussian fields related to the Breuer-Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.

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We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry's random wave model to a finite collection of (possibly overlapping) smooth compact subsets of R 2 . Our main result shows that, as the energy diverges to infinity and after an adequate normalisation, these random elements converge in distribution to a Gaussian vector, whose covariance structure reproduces that of a homogeneous independently scattered random measure. A by-product of our analysis is that, when restricted to rectangles, the dominant chaotic projection of the nodal length field weakly converges to a standard Wiener sheet, in the Banach space of real-valued continuous mappings over a fixed compact set. An analogous study is performed for complex-valued random waves, in which case the nodal set is a locally finite collection of random points.