The Gamma Stein equation and non-central de Jong theorems

in Bernoulli (in press)

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

Limit theorems for symmetric U-statistics using contractions

E-print/Working paper (2018)

Sojourn time dimensions of fractional Brownian motion

E-print/Working paper (2018)

A Stein deficit for the logarithmic Sobolev inequality

in Science China Mathematics (2017), 60

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 ▲]

Classical and free fourth moment theorems: universality and thresholds

in Journal of Theoretical Probability (2016), 29(2), 653-680

Strong asymptotic independence on Wiener chaos

in Proceedings of the American Mathematical Society (2016), 144(2), 875-886

Multidimensional limit theorems for homogeneous sums : a general transfer principle

in ESAIM: Probability and Statistics = Probabilité et statistique : P & S (2016), 20