Multidimensional limit theorems for homogeneous sums: a general transfer principle

TL;DR

This paper extends the fourth moment criterion to multidimensional homogeneous sums in both classical and free probability settings, enabling a transfer principle for the Central Limit Theorem across different types of random variables.

Contribution

It establishes a multidimensional version of the fourth moment criterion and extends the transfer principle for the CLT to broader classes of homogeneous sums.

Findings

Multidimensional fourth moment criterion proven for homogeneous sums.

Transfer principle for CLT extended to independent and free random variables.

Applicable to variables with zero third and non-negative fourth cumulant.

Abstract

The aim of the present paper is to establish the multidimensional counterpart of the \textit{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 \cite{NPPS} 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.