标题： Dirichlet过程泛函的多重积分表示 显示英文标题

标题： Multiple integral representation for functionals of Dirichlet processes

摘要： 我们指出，之前由作者引入的有限瓮序列对称统计量的Hoeffding-ANOVA分解的恰当使用，可以将Dirichlet-Ferguson过程（记作$L^2(D)$）的平方可积泛函空间分解为正交子空间，这些子空间的多重积分阶数递增。 这给出了$L^2(D)$和某个确定性函数类上的福克空间之间的同构关系。 通过Blackwell和MacQueen的一个著名结果，我们证明了$n$阶正交多重积分空间中的每个元素都可以表示为$L^2$阶渐近极限下的退化核度为$n$的$U$统计量。 给出了将给定泛函分解为线性组合形式的一般公式，其条件期望的系数明确计算。 我们表明，在简单情况下，多重积分可以用雅可比多项式表示。 建立了多个联系，特别是在贝叶斯决策问题方面，以及与Littler和Fackerell、Griffiths关于多等位基因扩散模型转移密度的经典公式有关的联系。 我们的结果还可用于通过可交换观测向量的有限维数的$U$统计量来计算$L^2(D)$元素的最佳逼近。 显示英文摘要

摘要： 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.