Abstract :
[en] Functionals of random fields have always been a central topic in probability theory,
since its inception as a subject of study. The latter include, among others, partial
sums of random variables and geometric quantities associated to random functions
on manifolds. In this thesis, we investigate the asymptotic probabilistic behaviour
of integral functionals of weakly stationary random fields on expanding Euclidean
domains, with a special focus on additive (or nonlinear) functionals of stationary
Gaussian fields.
In Chapter 1 we first introduce the main mathematical objects and tools encoun-
tered in this work, concluding with an overview of the state of the art and our new
contributions related to the main research questions of this thesis. The two main
questions are the following: first, as the integration domain expands, does a central
limit theorem hold? Second, given two expanding integration domains, what is the
asymptotic covariance between their integral functionals?
Chapter 2 contains the paper "Spectral central limit theorem for additive func-
tionals of isotropic and stationary Gaussian fields", written in collaboration with Ivan
Nourdin. In this chapter, we prove that a large class of additive functionals of station-
ary, isotropic Gaussian fields satisfies a central limit theorem if an easily verifiable
condition on the spectral measure holds. This result brings to light a new class of
"strongly correlated" Gaussian fields whose additive functionals satisfy a central limit
theorem. This fact contradicts the intuition forged in the last four decades, starting
from the seminal works by Breuer, Dobrushin, Major, Rosenblatt and Taqqu.
Chapter 3 contains the paper "Fluctuations of polyspectra in spherical and Eu-
clidean random wave models", written in collaboration with Francesco Grotto and
Anna Paola Todino. Our main result provides the variance rate of any additive func-
tional of Euclidean (Berry’s random wave model) and spherical random waves, a
problem that was left as a conjecture ten years ago. To do this, we exploit a relation
between random waves and Pearson’s random walks.
Chapter 4 contains the paper "Asymptotic covariances for functionals of weakly
stationary random fields". Here we compute the asymptotic covariances of integral
functionals of weakly stationary random fields on expanding domains under assump-
tions that encompass the ones in the literature, deriving an explicit formula that
involves the directional derivative of the cross covariogram of two domains.
Chapter 5 contains the preprint "Limit theorems for p-domain functionals of
stationary Gaussian fields", written in collaboration with Nikolai Leonenko, Ivan
Nourdin and Francesca Pistolato. In this chapter we consider more general families
of additive functionals, which we call p-domain functionals, including as a special
case spatio-temporal functionals and 1-domain functionals considered in the previous
chapters. In this setting, we are able (under suitable assumptions) to reduce the
study of p-domain functionals to that of some 1-domain functionals, explaining some
recent findings in the literature in a new light.
