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February 2008 Multiple integral representation for functionals of Dirichlet processes
Giovanni Peccati
Bernoulli 14(1): 91-124 (February 2008). DOI: 10.3150/07-BEJ5169


We point out that a proper use of the Hoeffding–ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet–Ferguson process, written $L^2(D)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^2(D)$ and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, 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 degenerate 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. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of $L^2(D)$ by means of $U$-statistics of finite vectors of exchangeable observations.


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Giovanni Peccati. "Multiple integral representation for functionals of Dirichlet processes." Bernoulli 14 (1) 91 - 124, February 2008.


Published: February 2008
First available in Project Euclid: 8 February 2008

zbMATH: 1175.60072
MathSciNet: MR2401655
Digital Object Identifier: 10.3150/07-BEJ5169

Rights: Copyright © 2008 Bernoulli Society for Mathematical Statistics and Probability


Vol.14 • No. 1 • February 2008
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