Vector-valued semicircular limits on the free Poisson chaos

DOI10.1214/16-ECP12zbMATH Open1362.46065arXiv1512.00494MaRDI QIDQ321141

Solesne Bourguin

Publication date: 7 October 2016

Published in: Electronic Communications in Probability (Search for Journal in Brave)

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).

Full work available at URL: https://arxiv.org/abs/1512.00494

zbMATH Keywords

fourth moment theoremdiagram formulafree Poisson chaosmultidimensional free limit theoremssemicircular distribution

Mathematics Subject Classification ID

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

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