Electronic Communications in Probability

Gaussian Approximations of Multiple Integrals

Giovanni Peccati

Full-text: Open access

Abstract

Fix $k\geq 1$, and let $I(l), l \geq 1$, be a sequence of $k$-dimensional vectors of multiple Wiener-Itô integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as $l \to\infty$, the law of $I(l)$ is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a $k$-dimensional Gaussian vector having the same covariance matrix as $I(l)$. The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of $I(l)$. In particular, we will not assume that the covariance matrix of $I(l)$ is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 34, 350-364.

Dates
Accepted: 13 October 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224977

Digital Object Identifier
doi:10.1214/ECP.v12-1322

Mathematical Reviews number (MathSciNet)
MR2350573

Zentralblatt MATH identifier
1130.60029

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Gaussian processes Malliavin calculus Multiple stochastic integrals Non-central limit theorems Weak convergence

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Peccati, Giovanni. Gaussian Approximations of Multiple Integrals. Electron. Commun. Probab. 12 (2007), paper no. 34, 350--364. doi:10.1214/ECP.v12-1322. https://projecteuclid.org/euclid.ecp/1465224977


Export citation

References

  • P. Baldi and D. Marinucci (2007). Some characterizations of the spherical harmonics coefficients for isotropic random fields. Statistics and Probability Letters 77(5), 490-496.
  • J.M. Corcuera, D. Nualart and J.H.C. Woerner (2006). Power variation of some integral long memory process. Bernoulli 12(4), 713-735.
  • P. Deheuvels, G. Peccati and M. Yor (2006). On quadratic functionals of the Brownian sheet and related processes. Stochastic Processes and their Applications 116, 493-538.
  • R.M. Dudley (2003). Real Analysis and Probability. (2nd Edition). Cambridge University Press, Cambridge.
  • Y. Hu and D. Nualart (2005). Renormalized self-intersection local time for fractional Brownian motion. The Annals of Probabability 33(3), 948-983.
  • S. Janson (1997). Gaussian Hilbert Spaces. Cambridge University Press, Cambridge.
  • D. Marinucci (2006) High-resolution asymptotics for the angular bispectrum of spherical random fields. The Annals of Statistics 34, 1-41.
  • D. Marinucci (2007). A Central Limit Theorem and Higher Order Results for the Angular Bispectrum. To appear in: Probability Theory and Related Fields.
  • D. Marinucci and G. Peccati (2007a). High-frequency asymptotics for subordinated stationary fields on an Abelian compact group. To appear in: Stochastic Processes and their Applications.
  • D. Marinucci and G. Peccati (2007b). Group representation and high-frequency central limit theorems for subordinated spherical random fields. Preprint. math.PR/0706.2851v3
  • A. Neuenkirch and I. Nourdin (2006). Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. To appear in: Journal of Theoretical Probability.
  • D. Nualart (2006). The Malliavin Calculus and Related Topics (2nd Edition). Springer-Verlag. Berlin-Heidelberg-New York.
  • D. Nualart and S. Ortiz-Latorre (2007). Central limit theorems for multiple stochastic integrals and Malliavin calculus. To appear in: Stochastic Processes and their Applications.
  • D. Nualart and G. Peccati (2005). Central limit theorems for sequences of multiple stochastic integrals. The Annals of Probability 33, 177-193.
  • G. Peccati and C.A. Tudor (2005). Gaussian limits for vector-valued multiple stochastic integrals. In: Séminaire de Probabilités XXXVIII, LNM 1857, 247-262, Springer-Verlag. Berlin-Heidelberg-New York.
  • D. Revuz and M. Yor (1999). Continuous Martingales and Brownian Motion. Springer-Verlag Berlin-Heidelberg-New York.
  • D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskiui (1988). Quantum theory of angular momentum.Irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols (Translated from the Russian). World Scientific Publishing Co., Inc., Teaneck, NJ..