The determinant of the Malliavin matrix and the determinant of the covariance matrix for multiple integrals
Résumé
A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order, we show that the expectation of the determinant of the Malliavin matrix is bigger than the determinant of the covariance matrix multiplied by a positive constant. As a consequence we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional and that the result is not true for random variables in Wiener chaoses of different orders. We will also relax a condition in the Fourth Moment Theorem proven by Nualart and Peccati in \cite{NP}.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)