CS09: Limit theorems through the lens of Wiener chaos and Stein-Malliavin Techniques

date: 7/15/2025, time: 16:00-17:30, room: IM EM

Organizer: Anna Paola Todino (University of Eastern Piedmont) & & Claudio Durastanti (Sapienza University of Rome)

Chair: Claudio Durastanti (Sapienza Università di Roma)

Functional second-order Gaussian Poincaré inequalities

Anna Vidotto (Sapienza University of Rome)

In this talk, we operate in the framework of Hilbert-valued Wiener structures and derive a functional version of the second-order Gaussian Poincaré inequality that leads to abstract bounds for Gaussian process approximation in \(d_2\) distance. Our abstract bounds are flexible and can be applied in various examples including functional Breuer-Major central limit theorems, shallow neural networks, and spatial statistics of SPDE solutions. Based on a joint work with Guangqu Zheng (Boston University).

The discrepancy between min-max statistics of Gaussian and Gaussian-subordinated matrices

Nicola Turchi (Università degli Studi di Milano-Bicocca)

We compute quantitative bounds for measuring the discrepancy between the distributions of two min-max statistics involving either pairs of Gaussian random matrices, or one Gaussian and one Gaussiansubordinated random matrix. In the fully Gaussian setup, our approach allows us to recover quantitative versions of well-known inequalities by Gordon, thus generalising the quantitative version of the Sudakov-Fernique inequality deduced by Chatterjee. On the other hand, the Gaussian-subordinated case yields generalizations of estimates obtained in the framework of the Chernozhukov-Chetverikov-Kato (CCK) theory. As applications, we establish comparison bounds for order statistics of random vectors and fourth moment bounds for matrices of multiple stochastic Wiener-Itô integrals.

Lipschitz-Killing curvatures for excursion sets of spin spherical random fields

Francesca Pistolato (Université du Luxembourg)

In the present era, there is a growing interest in modeling data not only with scalar values but also with more sophisticated algebraic structures. An important example are spin spherical random fields, that can be defined as random sections of a bundle of the 2-sphere. These fields manifest in both gravitational lensing data and Cosmic Microwave Background polarization data, which can be seen as vectors on the complex plane. Motivated by studying anisotropies and divergence from Gaussianity of the latter, we study the excursion sets of their real part by means of their Lipschitz-Killing curvatures, that are geometric functionals such as, in dimension 3, the volume, the surface area, the cross-sectional diameter and the Euler characteristic. Without requiring the isotropy of the field, it is possible to compute explicitly the expectation of such functionals in terms of the spin parameter and the level of the excursion. Moreover, it is possible to explicit a chaotic decomposition for the second curvature, the surface area, providing a small step forward in the study of second-order asymptotics of such functionals in the high-frequency regime. The talk is based on joint works with M. Stecconi.