Electronic Communications in Probability

Convergence in law in the second Wiener/Wigner chaos

Ivan Nourdin and Guillaume Poly

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Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset $L_0$ of $L$ satisfying that, for any $F_\infty$ in $L_0$, the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to $F_\infty$. This result is in the spirit of the seminal paper by Nualart and Peccati, in which the authors discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we offer analogues of this result in the case of free Brownian motion and double Wigner integrals, in the context of free probability.

An Erratum is available in ECP volume 17 paper number 54.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 36, 12 pp.

Accepted: 18 August 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras
Secondary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes 60H05: Stochastic integrals

Convergence in law second Wiener chaos second Wigner chaos quadratic form free probability

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Nourdin, Ivan; Poly, Guillaume. Convergence in law in the second Wiener/Wigner chaos. Electron. Commun. Probab. 17 (2012), paper no. 36, 12 pp. doi:10.1214/ECP.v17-2023. https://projecteuclid.org/euclid.ecp/1465263169

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  • Biane, Philippe; Speicher, Roland. Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields 112 (1998), no. 3, 373–409.
  • Deya, Aurélien; Nourdin, Ivan. Convergence of Wigner integrals to the tetilla law. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012), 101–127.
  • Hellerstein, S.; Korevaar, J. Limits of entire functions whose growth and zeros are restricted. Duke Math. J. 30 1963 221–227.
  • Kemp, T., Nourdin, I., Peccati, G. and Speicher, R.: Wigner chaos and the fourth moment. Ann. Probab. 40, (2012), no. 4, 1577-1635.
  • Nica, Alexandru; Speicher, Roland. Lectures on the combinatorics of free probability. London Mathematical Society Lecture Note Series, 335. Cambridge University Press, Cambridge, 2006. xvi+417 pp. ISBN: 978-0-521-85852-6; 0-521-85852-6
  • Nourdin, I.: Lectures on Gaussian approximations with Malliavin calculus. Séminaire de Probabilités, to appear.
  • Nourdin, Ivan; Peccati, Giovanni. Noncentral convergence of multiple integrals. Ann. Probab. 37 (2009), no. 4, 1412–1426.
  • Nourdin, I. and Peccati, G.: Poisson approximations on the free Wigner chaos. Preprint (2011).
  • Nourdin, I. and Peccati, G.: Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality. Cambridge Tracts in Mathematics. Cambridge University Press, (2012).
  • Nualart, David; Peccati, Giovanni. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005), no. 1, 177–193.
  • Schreiber, Michel. Fermeture en probabilité de certains sous-espaces d'un espace L2. Application aux chaos de Wiener. (French) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 14 (1969/70), 36–48.
  • Sevastyanov, B.A.: A class of limit distributions for quadratic forms of normal stochastic variables. Theor. Probab. Appl. 6, (1961), 337-340.