Vector-valued semicircular limits on the free Poisson chaos

TL;DR

This paper establishes a multidimensional central limit theorem for vectors of multiple integrals on the free Poisson chaos, showing that component-wise convergence implies joint convergence to the semicircular distribution.

Contribution

It extends the free Poisson chaos CLT to a multidimensional setting, demonstrating joint convergence from component-wise convergence, a novel result in free probability theory.

Findings

Component-wise convergence implies joint convergence to semicircular distribution.

The result parallels classical and other non-commutative chaos CLTs.

Enhances understanding of limit behaviors in free probability.

Abstract

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the free Poisson chaos is obtained as part of the limit theorem: component-wise convergence of sequences of vectors of multiple integrals with respect to a free Poisson random measure toward the semicircular distribution implies joint convergence. This result complements similar findings for the Wiener chaos by Peccati and Tudor (2005), the classical Poisson chaos by Peccati and Zheng (2010) and the Wigner chaos by Nourdin, Peccati and Speicher (2013).