Annals of Probability

Central limit theorems for sequences of multiple stochastic integrals

David Nualart and Giovanni Peccati

Full-text: Open access


We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting behavior of quadratic functionals of Gaussian processes.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 177-193.

First available in Project Euclid: 11 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60H05: Stochastic integrals

Multiple stochastic integrals Brownian motion weak convergence fractional Brownian motion Brownian sheet


Nualart, David; Peccati, Giovanni. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005), no. 1, 177--193. doi:10.1214/009117904000000621.

Export citation


  • Alòs, E. and Nualart, D. (2003). Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 129–152.
  • Deheuvels, P. and Martynov, G. (2003). Karhunen–Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions. In High Dimensional Probability III (J. Hoffman-Jørgensen, M. B. Marous and J. A. Wellner, eds.) 55 57–93. Birhäuser, Basel.
  • Deheuvels, P., Peccati, G. and Yor, M. (2004). Some identities in law for quadratic functionals of a Brownian sheet and its bridges. Preprint, Univ. Paris VI. Prépublication n. 910 du Laboratoire de Probabilités et Modèles Aléatoires, Univ. Paris VI et Univ. Paris VII.
  • Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 191–212.
  • Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Univ. Press.
  • Jeulin, T. (1980). Semimartingales et Grossissement d'une Filtration. Lecture Notes in Math. 833. Springer, Berlin.
  • Major, P. (1981). Multiple Wiener–Itô Integrals. Lecture Notes in Math. 849. Springer, New York.
  • Maruyama, G. (1982). Applications of the multiplication of the Itô–Wiener expansions to limit theorems. Proc. Japan Acad. 58 388–390.
  • Maruyama, G. (1985). Wiener functionals and probability limit theorems, I: The central limit theorem. Osaka J. Math. 22 697–732.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, Berlin.
  • Peccati, G. and Yor, M. (2004a). Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge. In Asymptotic Methods in Stochastics 49–74. Amer. Math. Soc., Providence, RI.
  • Peccati, G. and Yor, M. (2004b). Hardy's inequality in $ L^{2}( [ 0,1] ) $ and principal values of Brownian local times. In Asymptotic Methods in Stochastics 75–87. Amer. Math. Soc., Providence, RI.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • Stroock, D. W. (1987). Homogeneous chaos revisited. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247 1–8. Springer, Berlin.
  • Surgailis, D. (2000). CLTs for polynomials of linear sequences: Diagram formula with illustrations. In Long Range Dependence 111–128. Birkhäuser, Basel.
  • Üstünel, A. S. and Zakai, M. (1989). Independence and conditioning on Wiener space. Ann. Probab. 17 1441–1453.
  • Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5 67–85.