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

Abstract

The aim of the present paper is to establish the multidimensional counterpart of the fourth moment criterion for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in [22] in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

High-dimensional central limit theorems for homogeneous sums

“…For example, the universality results analogous to (II) have also been established for Poisson variables in Peccati & Zheng [57] and i.i.d. variables with zero skewness and non-negative excess kurtosis in Nourdin et al [49,50], respectively. Also, the recent work of Döbler & Peccati [28] has extended (I) and (III) to more general degenerate U -statistics which were originally treated in [22].…”

High-Dimensional Central Limit Theorems for Homogeneous Sums

“…For example, the universality results analogous to (II) have also been established for Poisson variables in Peccati & Zheng [52] and i.i.d. variables with zero skewness and nonnegative excess kurtosis in Nourdin et al [45,46], respectively. Also, the recent work of Döbler & Peccati [28] has extended (I) and (II) to more general degenerate U -statistics which were originally treated in [22].…”