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Berry-Esseen bounds in the Breuer-Major CLT and Gebelein's inequality Nourdin, Ivan ; Peccati, Giovanni ; Yang, Xiaochuan in Electronic Communications in Probability (2019), 24(34), 1-12 Detailed reference viewed: 127 (2 UL)Nodal Statistics of Planar Random Waves Nourdin, Ivan ; Peccati, Giovanni ; in Communications in Mathematical Physics (2019), 369(1), 99-151 Detailed reference viewed: 342 (139 UL)Quantitative CLTs for symmetric U-statistics using contractions Döbler, Christian ; Peccati, Giovanni in Electronic Journal of Probability (2019) Detailed reference viewed: 95 (3 UL)Quantitative limit theorems for local functionals of arithmetic random waves Peccati, Giovanni ; Rossi, Maurizia in Celledoni, Elena (Ed.) Computation and combinatorics in dynamics, stochastics and control. (2018) Detailed reference viewed: 60 (0 UL)Sojourn time dimensions of fractional Brownian motion Nourdin, Ivan ; Peccati, Giovanni ; E-print/Working paper (2018) Detailed reference viewed: 87 (11 UL)Fourth moment theorems on the Poisson space: analytic statements via product formulae Döbler, Christian ; Peccati, Giovanni in Electronic Communications in Probability (2018), 23 Detailed reference viewed: 134 (5 UL)New Kolmogorov bounds for functionals of binomial point processes Peccati, Giovanni ; in Annals of Applied Probability (2017), 27(4), 1992-20131 Detailed reference viewed: 108 (4 UL)Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner Formula ; Nourdin, Ivan ; Peccati, Giovanni 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 ▲] Detailed reference viewed: 186 (10 UL)A Stein deficit for the logarithmic Sobolev inequality ; Nourdin, Ivan ; Peccati, Giovanni in Science China Mathematics (2017), 60 Detailed reference viewed: 150 (4 UL)Quantitative de Jong theorems in any dimension Döbler, Christian ; Peccati, Giovanni in Electronic Journal of Probability (2017), 22 Detailed reference viewed: 271 (26 UL)Classical and free fourth moment theorems: universality and thresholds Nourdin, Ivan ; Peccati, Giovanni ; et al in Journal of Theoretical Probability (2016), 29(2), 653-680 Detailed reference viewed: 170 (11 UL)Stochastic analysis for Poisson point processes: Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry Peccati, Giovanni ; Book published by Springer (2016) Detailed reference viewed: 281 (5 UL)Squared chaotic random variables: new moment inequalities with applications ; Nourdin, Ivan ; Peccati, Giovanni et al in Journal of Functional Analysis (2016), 270(2), 649-670 Detailed reference viewed: 214 (18 UL)Multidimensional limit theorems for homogeneous sums : a general transfer principle Nourdin, Ivan ; Peccati, Giovanni ; et al in ESAIM: Probability and Statistics (2016), 20 Detailed reference viewed: 80 (2 UL)Non-universality of nodal length distribution for arithmetic random waves ; Peccati, Giovanni ; Rossi, Maurizia et al in Geometric and Functional Analysis (2016), 26(3), 926-960 Detailed reference viewed: 147 (21 UL)Strong asymptotic independence on Wiener chaos Nourdin, Ivan ; ; Peccati, Giovanni in Proceedings of the American Mathematical Society (2016), 144(2), 875-886 Detailed reference viewed: 210 (15 UL)Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization ; Peccati, Giovanni ; in Probability Theory and Related Fields (2016), 165(3), 667-723 Detailed reference viewed: 233 (16 UL)The law of iterated logarithm for subordinated Gaussian sequences: uniform Wasserstein bounds Azmoodeh, Ehsan ; Peccati, Giovanni ; in ALEA: Latin American Journal of Probability and Mathematical Statistics (2016), 13 Detailed reference viewed: 107 (2 UL)Gaussian approximations of nonlinear statistis on the sphere ; ; et al in Journal of Mathematical Analysis and Applications (2016), 436(2), 1121-1148 Detailed reference viewed: 130 (8 UL)Multivariate Gaussian approxi- mations on Markov chaoses Campese, Simon ; Nourdin, Ivan ; Peccati, Giovanni et al in Electronic Communications in Probability (2016), 21 Detailed reference viewed: 226 (8 UL) |
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