Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Gaussian limits for vector-valued multiple stochastic integrals

Auteur(s):

Code(s) de Classification MSC:

• 60F05 Central limit and other weak theorems
• 60H05 Stochastic integrals

Résumé: We establish necessary and sufficient conditions for a sequence of $d$% -dimensional vectors of multiple stochastic integrals $\mathbf{F}% _{d}^{k}=\left( F_{1}^{k},...,F_{d}^{k}\right)$, $k\geq 1$, to converge in distribution to a $d$-dimensional Gaussian vector $\mathbf{N}_{d}=\left( N_{1},...,N_{d}\right)$. In particular, we show that if the covariance structure of $\mathbf{F}_{d}^{k}$ converges to that of $\mathbf{N}_{d}$, then componentwise convergence implies joint convergence. These results extend to the multidimensional case the main theorem of Nualart and Peccati (2003).

Mots Clés: Multiple stochastic integrals ; Limit theorems ; Weak convergence ; Brownian motion.

Date: 2003-11-04

Prépublication numéro: PMA-861