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): **

**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