Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle
Nourdin, Ivan1  ; Peccati, Giovanni1  ; Poly, Guillaume2  ; Simone, Rosaria3
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

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 :
Accepté le :
DOI : 10.1051/ps/2016014
Classification : 60F17, 60F05, 46L54
Keywords: Fourth moment phenomenon, free probability, homogeneous sums, multidimensional limit theorems, wiener chaos, wigner chaos

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. 
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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

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