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Multi-dimensional Semicircular Limits on the Free Wigner Chaos

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Book cover Seminar on Stochastic Analysis, Random Fields and Applications VII

Part of the book series: Progress in Probability ((PRPR,volume 67))

Abstract

We show that, for sequences of vectors of multiple Wigner integrals with respect to a free Brownian motion, componentwise convergence to semicircular law is equivalent to joint convergence. This result extends to the free probability setting some findings by Peccati and Tudor (2005), and represents a multi-dimensional counterpart of a limit theorem inside the free Wigner chaos established by Kemp, Nourdin, Peccati and Speicher (2011).

Mathematics Subject Classification (2010). 46L54, 60H05, 60H07, 60H30.

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Author information

Authors and Affiliations

  1. IECL, Université de Lorraine, 70239, F-54506, Vandoeuvre-lès-Nancy cedex, France

    Ivan Nourdin

  2. UR en Mathématiques, Université du Luxembourg, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

    Giovanni Peccati

  3. Fachrichtung Mathematik, Universität des Saarlandes, 151150, D-66041, Saarbrücken, Germany

    Roland Speicher

Authors
  1. Ivan Nourdin
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  2. Giovanni Peccati
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  3. Roland Speicher
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Corresponding author

Correspondence to Ivan Nourdin .

Editor information

Editors and Affiliations

  1. Institut de Mathématiques (IMA) Chaire de Statistiques Appliquées (STAP), Ecole Polytechnique Fédérale de Lausanne Fac. Sciences de Base (FSB), Lausanne, Switzerland

    Robert C. Dalang

  2. Institut Elie Cartan, Université de Lorraine, Vandoeuvre-les-Nancy CX, France

    Marco Dozzi

  3. Unité de Mathématiques appliquées, ENSTA ParisTech, Palaiseau, France

    Francesco Russo

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Nourdin, I., Peccati, G., Speicher, R. (2013). Multi-dimensional Semicircular Limits on the Free Wigner Chaos. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VII. Progress in Probability, vol 67. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0545-2_10

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