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

Abstract

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.