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

### Multiple integral representation for functionals of Dirichlet processes

Auteur(s):

Code(s) de Classification MSC:

• 60G09 Exchangeability
• 60G99 None of the above but in this section

Résumé: We point out that a proper use of the Hoeffding-ANOVA decomposition for symmetric statistics of finite urn sequences, introduced in Peccati (2002), yields a decomposition of the space of square integrable functionals of a Dirichlet-Ferguson process, written $L^{2}\left( D\right)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^{2}\left( D\right)$ ad an appropriate Fock space over a class of deterministic functions. By means of a well known result due to Blackwell and MacQueen (1973), we show that each element of the $n$-th orthogonal space of multiple integrals can be represented as the $L^{2}$ limit of $U$-statistics with degenerated kernel of degree $n$. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations, whose coefficients are explicitly computed. Our results are used to calculate the best approximation of elements of $L^{2}\left( D\right)$, by means of $U$% -statistics of finite vectors of exchangeable observations.

Mots Clés: Dirichlet Process ; Multiple Integrals ; Orthogonality ; Hoeffding-ANOVA decompositions ; Urn sequences ; Exchangeability ; $U$%-Statistics

Date: 2002-07-05

Prépublication numéro: PMA-748

Pdf file : PMA-748.pdf