Multi-dimensional Semicircular Limits on the Free Wigner Chaos

Progress in Probability

Volumn 67, Issue , 2013, Pages 211-221

  Nourdin, Ivan ^a   Peccati, Giovanni ^b   Speicher, Roland ^c  

^a UNIVERSITÉ DE LORRAINE   (France)

^b UNIVERSITY OF LUXEMBOURG   (Luxembourg)

^c SAARLAND UNIVERSITY   (Germany)

Author keywords

Convergence in distribution; fourth moment condition; free Brownian motion; free probability; multi dimensional limit theorems; semicircular law; Wigner chaos

Indexed keywords

EID: 85145076159     PISSN: 10506977     EISSN: 22970428     Source Type: Book Series    
DOI: 10.1007/978-3-0348-0545-2_10     Document Type: Chapter

References (16)

1. P. Biane, Free hypercontractivity. Comm. Math. Phys., 184 (2) (1997), 457–474.
2. P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Prob. Theory Rel. Fields, 112 (1998), 373–409.
3. A. Deya and I. Nourdin, Convergence of Wigner integrals to the tetilla law. Alea,9 (2012), 101–127.
4. T. Kemp, I. Nourdin, G. Peccati, and R. Speicher, Wigner chaos and the fourth moment. Ann. Probab., 40 (4) (2012), 1577–1635.
5. S. Janson, Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, 129, Cambridge University Press, 1997.
6. A. Nica and R. Speicher, Commutators of free random variables. Duke Math. J., 92 (3) (1998), 553–592.
7. A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability. Lecture Notes of the London Mathematical Society, 335, Cambridge University Press, 2006.
8. I. Nourdin and G. Peccati, Non-central convergence of multiple integrals. Ann. Probab., 37 (4) (2009), 1412–1426.
9. I. Nourdin and G. Peccati, Stein’s method meets Malliavin calculus: a short survey with new estimates. InRecent Development in Stochastic Dynamics and Stochastic Analysis, World Scientific, 207–236, (2010).
10. I. Nourdin and G. Peccati, Normal Approximations using Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics, Cambridge University Press, 2012.
11. I. Nourdin and G. Peccati, Poisson approximations on the free Wigner chaos. Ann. Probab., to appear.
12. I. Nourdin, G. Peccati, and G. Reinert, Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab., 38 (5) (2010), 1947–1985.
13. D. Nualart, The Malliavin Calculus and Related Topics. Springer Verlag, Berlin, second edition, 2006.
14. D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab., 33 (1) (2005), 177–193.
15. G. Peccati and M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams. Springer-Verlag, 2010.
16. G. Peccati and C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. InSéminaire de Probabilités XXXVIII, 247–262, (2004).