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Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization ; Nourdin, Ivan ; Zheng, Guangqu et al in Probability and Mathematical Statistics (2018), 38(2), 271-286 Detailed reference viewed: 219 (12 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: 163 (10 UL)A Stein deficit for the logarithmic Sobolev inequality ; Nourdin, Ivan ; Peccati, Giovanni in Science China Mathematics (2017), 60 Detailed reference viewed: 121 (3 UL)Exchangeable pairs on Wiener chaos Nourdin, Ivan ; Zheng, Guangqu in High-Dimensional Probability VIII Proceedings (2017) Detailed reference viewed: 95 (5 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: 142 (8 UL)Quantitative stable limit theorems on the Wiener space Nourdin, Ivan ; ; Peccati, Giovanni in Annals of Probability (2016), 44(1), 1-41 Detailed reference viewed: 187 (12 UL)Asymptotic behaviour of the cross-variation of some integral long memory processes Nourdin, Ivan ; in Probability and Mathematical Statistics (2016), 36(1), Detailed reference viewed: 54 (7 UL)Fisher information and the Fourth Moment Theorem Nourdin, Ivan ; in Annales de l'Institut Henri Poincare (B) Probability & Statistics (2016), 52(2), 849-867 Detailed reference viewed: 247 (13 UL)Convergence in law implies convergence in total variation for polynomials in independent Gaussian, Gamma or Beta random variables Poly, Guillaume Joseph ; Nourdin, Ivan in Progress in Probability (2016), 71 Detailed reference viewed: 115 (4 UL)Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization ; Nourdin, Ivan ; Zheng, Guangqu et al in Probability and Mathematical Statistics (2016) Detailed reference viewed: 93 (6 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: 184 (16 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: 188 (15 UL)Multidimensional limit theorems for homogeneous sums : a general transfer principle Nourdin, Ivan ; Peccati, Giovanni ; et al in ESAIM: Probability and Statistics = Probabilité et statistique : P & S (2016), 20 Detailed reference viewed: 69 (1 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: 199 (8 UL)Stein's method, logarithmic Sobolev and transport inequalities ; Nourdin, Ivan ; Peccati, Giovanni in Geometric & Functional Analysis (2015), 25 Detailed reference viewed: 214 (16 UL)Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients ; ; et al E-print/Working paper (2015) Detailed reference viewed: 39 (0 UL)The optimal fourth moment theorem Nourdin, Ivan ; Peccati, Giovanni in Proceedings of the American Mathematical Society (2015), 143(7), 3123-3133 Detailed reference viewed: 240 (17 UL)Multivariate central limit theorems for averages of fractional Volterra processes and applications to parameter estimation Nourdin, Ivan ; ; Scientific journal (2015) Detailed reference viewed: 99 (9 UL)Comparison inequalities on the Wiener space Nourdin, Ivan ; Peccati, Giovanni ; in Stochastic Processes & Their Applications (2014), 124(4), 1566-1581 Detailed reference viewed: 95 (2 UL)Central and non-central limit theorems in a free probability setting Nourdin, Ivan ; in Journal of Theoretical Probability (2014), 27(1), 220-248 Detailed reference viewed: 84 (2 UL) |
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