The topic of the talk is the asymptotic behavior of integral functionals of Gaussian random fields, defined on non-uniformly growing domains, i.e. t_1D_1 x ... x t_pD_p, where D_i are compact sets in R^{d_i} and t_i >0, for any i = 1, ..., p. Assuming the separability of the covariance structure of the underlying Gaussian field, it is possible to prove limit theorems as every t_i grows, only knowing the behavior for a chosen index j. Collaterally, we partially answer a conjecture by Réveillac, Stauch and Tudor in 2012 and improve their estimate on the rate of convergence of the qth variation of the fractional Wiener sheet to a Gaussian distribution.

The first half of the talk will be devoted to stating the Fourth Moment Theorem proved by Nualart and Peccati in 2005, and its quantitative counterpart by Nourdin and Peccati four years later. Finally, we will apply these results to our object of interests, understanding why the separability assumption emerges naturally, yet proves unsatisfactory in some cases.

The talk is based on an ongoing project with N. N. Leonenko, L. Maini and I. Nourdin.