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See detailA Stein deficit for the logarithmic Sobolev inequality
Ledoux, Michel; Nourdin, Ivan UL; Peccati, Giovanni UL

in Science China Mathematics (in press)

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See detailGaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner Formula
Goldstein, Larry; Nourdin, Ivan UL; Peccati, Giovanni UL

in Annals of Applied Probability (in press)

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 ▲]

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See detailCorrector in random homogenization of elliptic equations in presence of long-range media
Lechiheb, Atef; Nourdin, Ivan UL; Zheng, Guangqu UL et al

in Probability and Mathematical Statistics (in press)

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See detailFreeness characterisations on free chaos spaces
Bourguin, Solesne UL; Nourdin, Ivan UL

E-print/Working paper (2017)

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See detailPhase singularities in complex arithmetic random waves
Dalmao, Federico; Nourdin, Ivan UL; Peccati, Giovanni UL et al

E-print/Working paper (2016)

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See detailClassical and free fourth moment theorems: universality and thresholds
Nourdin, Ivan UL; Peccati, Giovanni UL; Poly, Guillaume et al

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

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See detailAsymptotic behaviour of the cross-variation of some integral long memory processes
Nourdin, Ivan UL; Zintout, Rola

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

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See detailFisher information and the Fourth Moment Theorem
Nourdin, Ivan UL; Nualart, David

in Annales de l'Institut Henri Poincare (B) Probability & Statistics (2016), 52(2), 849-867

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See detailStrong asymptotic independence on Wiener chaos
Nourdin, Ivan UL; Nualart, David; Peccati, Giovanni UL

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

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See detailWeak symmetric integrals with respect to the fractional Brownian motion
Binotto, Giulia; Nourdin, Ivan UL; Nualart, David

E-print/Working paper (2016)

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See detailMultivariate Gaussian approxi- mations on Markov chaoses
Campese, Simon UL; Nourdin, Ivan UL; Peccati, Giovanni UL et al

in Electronic Communications in Probability (2016), 21

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See detailSquared chaotic random variables: new moment inequalities with applications
Malicet, Dominique; Nourdin, Ivan UL; Peccati, Giovanni UL et al

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

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See detailMultidimensional limit theorems for homogeneous sums : a general transfer principle
Nourdin, Ivan UL; Peccati, Giovanni UL; Poly, Guillaume et al

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

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See detailStein's method, logarithmic Sobolev and transport inequalities
Ledoux, Michel; Nourdin, Ivan UL; Peccati, Giovanni UL

in Geometric & Functional Analysis (2015), 25

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See detailThe optimal fourth moment theorem
Nourdin, Ivan UL; Peccati, Giovanni UL

in Proceedings of the American Mathematical Society (2015), 143(7), 3123-3133

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See detailQuantitative stable limit theorems on the Wiener space
Nourdin, Ivan UL; Nualart, David; Peccati, Giovanni UL

in Annals of Probability (2015)

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See detailMultivariate central limit theorems for averages of fractional Volterra processes and applications to parameter estimation
Nourdin, Ivan UL; Nualart, David; Zintout, Rola

Scientific journal (2015)

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See detailLocal universality of the number of zeros of random trigonometric polynomials with continuous coefficients
Azais, Jean-Marc; Dalmao, Federico; Leon, Jose et al

E-print/Working paper (2015)

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See detailEntropy and the fourth moment phenomenon
Nourdin, Ivan UL; Peccati, Giovanni UL; Swan, Yvik UL

in Journal of Functional Analysis (2014), 266(5), 3170-3207

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