Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- P. DEHEUVELS
- G. PECCATI
**M. YOR**

**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*