Spectral characterizations of central limit theorems for functionals of stationary Gaussian fields

Speakers: Leonardo Maini\n\nWe consider Fourier-Hermite functionals (at a fixed frequency) of stationary Gaussian fields indexed by locally compact Abelian groups. Under suitable assumptions, we prove that convergence to a Gaussian law is equivalent to the absence of dominant spectral partitions at the fixed frequency. The assumptions of the criterion always hold for functionals of Gaussian fields considered in the literature, in particular the assumptions are particularly easy to be checked when we focus on classical functionals of Gaussian fields indexed by the flat torus, Euclidean space or lattice. This characterization provides a common spectral interpretation of classical short/ long-memory results and recent results on random waves, and leads to applications to random trigonometric polynomials, arithmetic random waves, cyclical long memory models, singular spectral measures, Euclidean random waves, and periodograms. In particular, for Euclidean Gaussian fields with spectral measures uniformly distributed on hypersurfaces, such as the sphere in the case of Euclidean random waves, our criterion allows to prove CLTs under assumptions on the geometry of the hypersurface.\n\nhttps://indico.math.cnrs.fr/event/16891/