Abstract
Fix $k\geq 1$, and let $I(l), l \geq 1$, be a sequence of $k$-dimensional vectors of multiple Wiener-Ito 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.
Highlights
Let U (l) = (U1 (l), ..., Uk (l)), l ≥ 1, be a sequence of centered random observations with values in Rk
Where the integers d1, ..., dk ≥ 1 do not depend on l, Idj indicates a multiple stochastic integral of order dj, and each fl(j) ∈ H⊙dj, j = 1, ..., k, is a symmetric kernel
We shall prove that, whenever the elements of the vectors I (l) have bounded variances (and without any further requirements on the covariance matrix of I (l)), the following three conditions are equivalent as l → +∞: (i) γ (L (I (l)), L (N (l))) → 0, where L (·) indicates the law of a given random vector, N (l) is a Gaussian vector having the same covariance matrix as I (l), and γ is some appropriate metric on the space of probability measures on Rk; (ii) for every j = 1, ..., k, E
Summary
We shall prove that, whenever the elements of the vectors I (l) have bounded variances (and without any further requirements on the covariance matrix of I (l)), the following three conditions are equivalent as l → +∞:. (i) γ (L (I (l)) , L (N (l))) → 0, where L (·) indicates the law of a given random vector, N (l) is a Gaussian vector having the same covariance matrix as I (l), and γ is some appropriate metric on the space of probability measures on Rk;. Note that the results of this paper are a generalization of the following theorem, which combines results proved in [13], [14] and [15]. The techniques we use to achieve our main results are once again the DDS Theorem, combined with Burkholder-Davis-Gundy inequalities and some results (taken from [4, Section 11.7]) concerning ‘uniformities’ over classes of probability measures.
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