Multiple integral representation for functionals of Dirichlet processes

TL;DR

This paper develops a novel orthogonal decomposition of square-integrable functionals of Dirichlet processes using multiple integrals, linking them to Fock spaces, U-statistics, and classical polynomial representations.

Contribution

It introduces a new decomposition of $L^2(D)$ for Dirichlet processes, connecting multiple integrals, U-statistics, and classical polynomial representations, with explicit formulas and applications.

Findings

Decomposition of $L^2(D)$ into orthogonal subspaces of multiple integrals.

Representation of elements as limits of U-statistics with degenerate kernels.

Connections established with Bayesian problems and diffusion models.

Abstract

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.