Electronic Communications in Probability

Vector-valued semicircular limits on the free Poisson chaos

Solesne Bourguin

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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).

Article information

Electron. Commun. Probab. Volume 21 (2016), paper no. 55, 11 pp.

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

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Primary: 46L54: Free probability and free operator algebras 81S25: Quantum stochastic calculus 60H05: Stochastic integrals

fourth moment theorem diagram formula multidimensional free limit theorems semicircular distribution free Poisson chaos

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Bourguin, Solesne. Vector-valued semicircular limits on the free Poisson chaos. Electron. Commun. Probab. 21 (2016), paper no. 55, 11 pp. doi:10.1214/16-ECP12. https://projecteuclid.org/euclid.ecp/1472830237

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