Gaussian Approximations of Multiple Integrals

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.