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

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},
     journal = {ESAIM: Probability and Statistics},
     pages = {293--308},
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     volume = {20},
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2016014/}
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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/

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