Issue |
ESAIM: PS
Volume 20, 2016
|
|
---|---|---|
Page(s) | 293 - 308 | |
DOI | https://doi.org/10.1051/ps/2016014 | |
Published online | 05 August 2016 |
Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle
1 Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, UR en Mathématiques, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg.
ivan.nourdin@uni.lu; giovanni.peccati@uni.lu
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 .
guillaume.poly@univ-rennes1.fr
3 Dipartimento di Scienze Politiche, Università degli Studi di Napoli − Federico II, Via L. Rodinò 22, 80138 Napoli, Italy.
rosaria.simone@unina.it
Received: 24 June 2015
Revised: 15 February 2016
Accepted: 19 May 2016
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.
Mathematics Subject Classification: 60F17 / 60F05 / 46L54
Key words: Fourth moment phenomenon / free probability / homogeneous sums / multidimensional limit theorems / wiener chaos / wigner chaos
© EDP Sciences, SMAI, 2016
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