Limit theorems for Gaussian fields via Chaos Expansions and Applications

In this PhD thesis, we apply a combination of Malliavin calculus and Stein's method in the framework of probability approximations. The specific problems we tackle with these methods are motivated by probabilistic models in cosmology (Part I: Quantitative CLTs for non linear functionals of random hyperspherical harmonics) and finance (Part II: The fractional Ornstein-Uhlenbeck process in rough volatility modelling). In this second part we also apply techniques from Large Deviations theory (Section: Short-time asymptotics for non self-similar stochastic volatility models).