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Vector-valued semicircular limits on the free Poisson chaos. (English) Zbl 1362.46065
Summary: In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by S. Bourguin and G. Peccati [J. Funct. Anal. 267, No. 4, 963–997 (2014; Zbl 1308.46071)]. 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 G. Peccati and C. A. Tudor [Lect. Notes Math. 1857, 247–262 (2005; Zbl 1063.60027)], the classical Poisson chaos by P. Giovanni and C. Zheng [Electron. J. Probab. 15, Paper No. 48, 1487–1527 (2010; Zbl 1228.60031)] and the Wigner chaos by I. Nourdin et al. [Prog. Probab. 67, 211–221 (2013; Zbl 1295.46046)].
MSC:
46L54 Free probability and free operator algebras
81S25 Quantum stochastic calculus
60H05 Stochastic integrals
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