Statistical inference for Vasicek-type model driven by Hermite processes

in Stochastic Processes and Their Applications (2019), 129(10), 3774-3791

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

Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization

in Probability and Mathematical Statistics (2018), 38(2), 271-286

Exchangeable pairs on Wiener chaos

in High-Dimensional Probability VIII Proceedings (2017)

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 ▲]

Convergence in law implies convergence in total variation for polynomials in independent Gaussian, Gamma or Beta random variables

in Progress in Probability (2016), 71

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

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

Strong asymptotic independence on Wiener chaos

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

Squared chaotic random variables: new moment inequalities with applications

in Journal of Functional Analysis (2016), 270(2), 649-670