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

### On quadratic functionals of the Brownian sheet and related processes

Auteur(s):

Code(s) de Classification MSC:

• 60F05 Central limit and other weak theorems
• 60F15 Strong theorems
• 60G15 Gaussian processes
• 60H07 Stochastic calculus of variations and the Malliavin calculus
• 62G30 Order statistics; empirical distribution functions

Résumé: Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on $\left[ 0,1\right] ^{2}$. In particular: (i) we use Fubini type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of Karhunen-Loève expansions, (iii) we prove central and non-central limit theorems in the same spirit of Peccati and Yor (2004) and Nualart and Peccati (2004). Our results extend some classical computations due to P. Lévy (1950), as well as the formulae recently obtained by Deheuvels and Martynov (2003).

Mots Clés: Empirical processes ; quadratic functionals ; Fubini argument

Date: 2004-05-10

Prépublication numéro: PMA-910