Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

Nourdin, Ivan; Peccati, Giovanni; Poly, Guillaume; Simone, Rosaria

doi:10.1051/ps/2016014

Nourdin, Ivan ¹ ; Peccati, Giovanni ¹ ; Poly, Guillaume ² ; Simone, Rosaria ³

¹ Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, UR en Mathématiques, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg.
² Unité de Formation et de Recherche Mathématiques MATH, Institut de Recherche Mathématiques de Rennes (IRMAR), 263 Avenue du General Leclerc, Building: B022, CS 74205, 35042 Rennes cedex, France .
³ Dipartimento di Scienze Politiche, Università degli Studi di Napoli − Federico II, Via L. Rodinò 22, 80138 Napoli, Italy.

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 293-308.

Résumé

We provide a synthetic yet comprehensive review of the so-called fourth moment criterion, and of universal limit theorems, for multilinear homogeneous sums, in both the classical and the free probability settings. In addition to such a general picture, we also prove a novel multidimensional transfer principle for Central Limit Theorems involving homogeneous sums with leptokurtic or mesokurtic entries. The key step will be to prove that joint and component-wise convergence are indeed equivalent for these random objects, encompassing well-known results concerning Wiener and Wigner Chaoses.

Reçu le : 2015-06-24
Accepté le : 2016-05-19

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/ps/2016014

Classification : 60F17, 60F05, 46L54
Mots-clés : Fourth moment phenomenon, free probability, homogeneous sums, multidimensional limit theorems, wiener chaos, wigner chaos

Affiliations des auteurs :

Nourdin, Ivan ¹ ; Peccati, Giovanni ¹ ; Poly, Guillaume ² ; Simone, Rosaria ³

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     title = {Multidimensional limit theorems for homogeneous sums: {A} survey and a general transfer principle},
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Nourdin, Ivan; Peccati, Giovanni; Poly, Guillaume; Simone, Rosaria. Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 293-308. doi : 10.1051/ps/2016014. http://www.numdam.org/articles/10.1051/ps/2016014/

Bibliographie
Cité par

M. Anshelevich, S.T. Belinschi, M. Bozejko and F. Lehner, Free infinite divisibility for $q$ -Gaussians. Math. Res. Lett. 17 (2010) 905–916. | DOI | MR | Zbl

O. Arizmendi, Convergence of the fourth moment and infinite divisibility. Probab. Math. Stat. 33 (2013) 201–212. | MR | Zbl

O. Arizmendi and A. Jaramillo, Convergence of the fourth moment and infinite divisibility: quantitative estimates. Electron. Commun. Probab. 19 (2014) 1–12. | DOI | MR | Zbl

O. Arizmendi and V. Pérez-Abreu, On the non-classical infinite divisibility of power semicircle distributions. Commun. Stochastic Anal. 4 (2010) 161–178. | MR | Zbl

E. Azmoodeh, S. Campese and G. Poly, Fourth Moment Theorems for Markov Diffusion Generators. J. Funct. Anal. 266 (2014) 2341–2359. | DOI | MR | Zbl

E. Azmoodeh, D. Malicet, G. Mijoule and G. Poly, Generalization of the Nualart-Peccati criterion. Ann. Probab. 44 (2016) 924–954. | DOI | MR | Zbl

B. Bhattacharya, P. Diaconis and S. Mukherjee, Universal limit theorems in graph coloring problems with connections to extremal combinatorics. Preprint (2014). | arXiv | MR

P. Biane and R. Speicher, Stochastic analysis with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Relat. Fields 112 (1998) 373–409. | DOI | MR | Zbl

S. Bourguin, Poisson convergence on the free Poisson algebra. Bernoulli 21 (2015) 2139–2156. | DOI | MR | Zbl

S. Bourguin and G. Peccati, Semicircular limits on the free Poisson chaos: counterexamples to a transfer principle. J. Funct. Anal. 267 (2013) 963–997. | DOI | MR | Zbl

S. Campese, I. Nourdin, G. Peccati and G. Poly, Multivariate Gaussian approximation on Markov chaoses. Preprint (2015). | arXiv | MR

L.H.Y. Chen, Stein meets Malliavin in Normal Approximation. Acta Math. Vietnamica 40 (2015) 205. | DOI | MR | Zbl

L.H.Y. Chen and G. Poly, Stein’s method, Malliavin calculus, Dirichlet forms and the fourth moment theorem. Festschrift Masatoshi Fukushima. Edited by Z.-Q. Chen, N. Jacob, M. Takeda and T. Uemura. In vol. 17 of Interdiscipl. Math. Sci. (2015) 107–130. | MR

P. De Jong, A central limit theorem for generalized multilinear forms. J. Multivar. Anal. 34 (1987) 275–289. | DOI | MR | Zbl

A. Deya and I. Nourdin, Convergence of Wigner integrals to the Tetilla law. ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2012) 101–127. | MR | Zbl

A. Deya and I. Nourdin, Invariance principles for homogeneous sums of free random variables. Bernoulli 20 (2013) 586–603. | MR | Zbl

A. Deya, S. Norredine and I. Nourdin, Fourth Moment Theorem and $q$ -Brownian Motion. Commun. Math. Phys. 321 (2013) 113–134. | DOI | MR | Zbl

P. Eichelsbacher and C. Thäle, New Berry-Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab. 19 (2014) 1–25. | DOI | MR | Zbl

T. Fissler and C. Thäle, A four moment theorem for Gamma limits on a Poisson chaos. Preprint (2015). | arXiv | MR

T. Kemp, I. Nourdin, G. Peccati and R. Speicher, Wigner Chaos and the fourth moment. Ann. Probab. 40 (2011) 1577–1635. | MR | Zbl

R. Lachiéze-Rey and G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs. Electron. J. Probab. 18 (2013) 1–32. | DOI | MR | Zbl

E. Mossel, Gaussian bounds for noise correlation of functions. Geometric Funct. Anal. 19 (2010) 1713–1756. | DOI | MR | Zbl

E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171 (2010) 295–341. | DOI | MR | Zbl

A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability. Cambridge University Press (1990). | MR | Zbl

I. Nourdin and G. Peccati, Noncentral convergence of multiple integrals. Ann. Probab. 37 (2009) 1412–1426. | DOI | MR | Zbl

I. Nourdin and G. Peccati, Normal approximations with Malliavin calculus: from Stein’s method to universality. Cambridge Tracts in Mathematics. Cambridge University Press (2012). | MR | Zbl

I. Nourdin and G. Peccati. Poisson approximations on the free Wigner chaos. Ann. Probab. 41 (2013) 2709–2723. | DOI | MR | Zbl

I. Nourdin, G. Peccati and G. Reinert, Invariance principles for homogeneous sums: universality of Gaussian Wiener Chaos. Ann. Probab. 38 (2010) 1947–1985. | DOI | MR | Zbl

I. Nourdin, G. Peccati and A. Reveillac, Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré, Probab. Stat. 46 (2010) 45–58. | DOI | Numdam | MR | Zbl

I. Nourdin, G. Peccati and R. Speicher, Multidimensional semicircular limits on the free Wigner Chaos. Ascona Proceedings 2011. In vol. 67 of Progress in Probability (2013) 211–221. | MR | Zbl

I. Nourdin, G. Peccati and Y. Swan, Entropy and the fourth moment phenomenon. J. Funct. Anal. 266 (2014) 3170–3207. | DOI | MR | Zbl

I. Nourdin, G. Peccati, G. Poly and R. Simone, Classical and free fourth moment theorems: universality and thresholds. J. Theoret. Probab. 29 (2016) 653–680. | DOI | MR | Zbl

D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005) 177–193. | DOI | MR | Zbl

G. Peccati, Quantitative CLTs on a Gaussian space: a survey of recent developments. ESAIM: Proc. Suv. 44 (2014) 61–78. | DOI | MR | Zbl

G. Peccati and M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams. Springer-Verlag (2010). | MR | Zbl

G. Peccati and Ch. Thaele, Gamma limits and U-statistics on the Poisson space. ALEA, Lat. Am. J. Probab. Math. Stat. 10 (2013) 525–560. | MR | Zbl

G. Peccati and C. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. In vol. XXXVIII, Séminaire de Probabilités (2005) 247–262. | MR | Zbl

G. Peccati and C. Zheng, Universal Gaussian fluctuations on the discrete Poisson chaos. Bernoulli 20 (2013) 697–715. | MR | Zbl

G. Peccati, J.L. Solé, M.S. Taqqu and F. Utzet, Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38 (2010) 443–478. | DOI | MR | Zbl

R. Simone, Universality for free homogeneous sums in every dimension. ALEA, Lat. Am. J. Probab. Math. Stat. 12 (2015) 213–244. | MR | Zbl

D. Voiculescu, Symmetries of some reduced free product $C^{*}$ -algebras. Operator algebras and their connection with topology and ergodic theory. Vol. 1132 of Lect. Notes Math. Springer (1985) 556–588. | MR | Zbl

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