Open Access
2016 Vector-valued semicircular limits on the free Poisson chaos
Solesne Bourguin
Electron. Commun. Probab. 21: 1-11 (2016). DOI: 10.1214/16-ECP12


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).


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Solesne Bourguin. "Vector-valued semicircular limits on the free Poisson chaos." Electron. Commun. Probab. 21 1 - 11, 2016.


Received: 29 March 2016; Accepted: 1 August 2016; Published: 2016
First available in Project Euclid: 2 September 2016

zbMATH: 1362.46065
MathSciNet: MR3548767
Digital Object Identifier: 10.1214/16-ECP12

Primary: 46L54 , 60H05 , 81S25

Keywords: diagram formula , Fourth moment theorem , free Poisson chaos , multidimensional free limit theorems , semicircular distribution

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