2007/07/09 by Giovanni Peccati, Peccati, Giovanni
Mathematics · #60B15 #60F05 #60G60 #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #math.PR #msc:60B15 #msc:60F05 #msc:60G60
paper · pdf · doi:10.48550/arxiv.0707.1220
15 pages
arxiv created 2007/07/09 · openalex publication_date 2007/07/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix an integer k, and let I(l), l=1,2,..., 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 diverges, 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.