Hoeffding decompositions for exchangeable sequences and chaotic representation of functionals of Dirichlet processes
Giovanni Peccati · Comptes Rendus Mathématique · 2003
Consider an exchangeable sequence X = { X n : 1 ⩽ n < N } , where N ∈ ℕ ∪ { ∞ } , and note 𝐗 n = ( X 1 , ... , X n ) . We say that X is Hoeffding decomposable if, for each n, every square integrable, centered and symmetric functional of X n is the orthogonal sum of n U-statistics with degenerated and symmetric kernels. We state a necessary and sufficient condition for an exchangeable sequence to be Hoeffding decomposable, named weak independence . We show that a class of weakly independent sequences is given by generalized urn sequences and, specifically, by generalized Pólya urns. We point out that this yields an orthogonal decomposition of the space of square integrable functionals of Dirichlet–Ferguson processes into orthogonal subspaces of multiple integrals. Explicit formulae are provided.