Nodal Statistics of Planar Random Waves

in Communications in Mathematical Physics (2019), 369(1), 99-151

Phase singularities in complex arithmetic random waves

in Electronic Journal of Probability (2019), 24(71), 1-45

Berry-Esseen bounds in the Breuer-Major CLT and Gebelein's inequality

in Electronic Communications in Probability (2019), 24(34), 1-12

Concentration of the Intrinsic Volumes of a Convex Body

in Geometric Aspects of Functional Analysis – Israel Seminar (GAFA) 2017-2019 (2018)

Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner Formula

in Annals of Applied Probability (2017), 27(1), 1-47

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical ... [more ▼]

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the present paper is to provide a Berry-Esseen bound for the normal approximation of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincar\'e inequality, (3) concentration estimates, and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy and Tropp (2014) about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets. [less ▲]

A Stein deficit for the logarithmic Sobolev inequality

in Science China Mathematics (2017), 60

Multidimensional limit theorems for homogeneous sums : a general transfer principle

in ESAIM: Probability and Statistics (2016), 20

Asymptotic behaviour of the cross-variation of some integral long memory processes

in Probability and Mathematical Statistics (2016), 36(1),

Fisher information and the Fourth Moment Theorem

in Annales de l'Institut Henri Poincare (B) Probability & Statistics (2016), 52(2), 849-867

Multivariate Gaussian approxi- mations on Markov chaoses

in Electronic Communications in Probability (2016), 21