Electronic Communications in Probability

Gaussian Approximations of Multiple Integrals

Giovanni Peccati

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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

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