Results 21-40 of 106.     A Stein deficit for the logarithmic Sobolev inequalityLedoux, Michel; Nourdin, Ivan ; Peccati, Giovanni in Science China Mathematics (2017), 60Detailed reference viewed: 112 (3 UL) Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner FormulaGoldstein, Larry; Nourdin, Ivan ; Peccati, Giovanni in Annals of Applied Probability (2017), 27(1), 1-47Intrinsic 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: 157 (8 UL) Classical and free fourth moment theorems: universality and thresholdsNourdin, Ivan ; Peccati, Giovanni ; Poly, Guillaume et alin Journal of Theoretical Probability (2016), 29(2), 653-680Detailed reference viewed: 138 (8 UL) Quantitative stable limit theorems on the Wiener spaceNourdin, Ivan ; Nualart, David; Peccati, Giovanni in Annals of Probability (2016), 44(1), 1-41Detailed reference viewed: 180 (12 UL) The law of iterated logarithm for subordinated Gaussian sequences: uniform Wasserstein boundsAzmoodeh, Ehsan ; Peccati, Giovanni ; Poly, Guillaumein ALEA: Latin American Journal of Probability and Mathematical Statistics (2016), 13Detailed reference viewed: 79 (2 UL) Stochastic analysis for Poisson point processes: Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometryPeccati, Giovanni ; Reitzner, MatthiasBook published by Springer (2016)Detailed reference viewed: 233 (3 UL) Strong asymptotic independence on Wiener chaosNourdin, Ivan ; Nualart, David; Peccati, Giovanni in Proceedings of the American Mathematical Society (2016), 144(2), 875-886Detailed reference viewed: 178 (15 UL) Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilizationLast, Guenter; Peccati, Giovanni ; Schulte, Matthiasin Probability Theory & Related Fields (2016), 165(3), 667-723Detailed reference viewed: 178 (15 UL) Multivariate Gaussian approxi- mations on Markov chaosesCampese, Simon ; Nourdin, Ivan ; Peccati, Giovanni et alin Electronic Communications in Probability (2016), 21Detailed reference viewed: 193 (8 UL) Squared chaotic random variables: new moment inequalities with applicationsMalicet, Dominique; Nourdin, Ivan ; Peccati, Giovanni et alin Journal of Functional Analysis (2016), 270(2), 649-670Detailed reference viewed: 168 (9 UL) Non-universality of nodal length distribution for arithmetic random wavesMarinucci, Domenico; Peccati, Giovanni ; Rossi, Maurizia et alin Geometric & Functional Analysis (2016), 26(3), 926-960Detailed reference viewed: 113 (18 UL) Multidimensional limit theorems for homogeneous sums : a general transfer principleNourdin, Ivan ; Peccati, Giovanni ; Poly, Guillaume et alin ESAIM: Probability and Statistics = Probabilité et statistique : P & S (2016), 20Detailed reference viewed: 67 (1 UL) Gaussian approximations of nonlinear statistis on the sphereBourguin, Solesne; Durastanti, Claudio; Marinucci, Domenico et alin Journal of Mathematical Analysis and Applications (2016), 436(2), 1121-1148Detailed reference viewed: 114 (8 UL) Stein's method, logarithmic Sobolev and transport inequalitiesLedoux, Michel; Nourdin, Ivan ; Peccati, Giovanni in Geometric & Functional Analysis (2015), 25Detailed reference viewed: 199 (16 UL) Optimal Berry-Esseen bounds on the Poisson space.Peccati, Giovanni ; Azmoodeh, Ehsan E-print/Working paper (2015)Detailed reference viewed: 37 (0 UL) The optimal fourth moment theoremNourdin, Ivan ; Peccati, Giovanni in Proceedings of the American Mathematical Society (2015), 143(7), 3123-3133Detailed reference viewed: 232 (17 UL) Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach.Azmoodeh, Ehsan ; Peccati, Giovanni ; Poly, Guillaumein Séminaire de Probabilités XLVII (2015)Detailed reference viewed: 125 (10 UL) Quantitative CLTs on a Gaussian space: a survey of recent developmentsPeccati, Giovanni in ESAIM: Proceedings and Surveys (2014), 44Detailed reference viewed: 84 (2 UL) Entropy and the fourth moment phenomenonNourdin, Ivan ; Peccati, Giovanni ; Swan, Yvik in Journal of Functional Analysis (2014), 266(5), 3170-3207Detailed reference viewed: 156 (11 UL) Universal Gaussian fluctuations on the discrete Poisson chaosPeccati, Giovanni ; Zheng, Cengboin Bernoulli (2014), 20(2), 697-715Detailed reference viewed: 79 (0 UL)