Hoeffding-anova Decompositions for Symmetric Statistics of Exchangeable Observations

Consider a (possibly infinite) exchangeable sequence X= {Xn : 1≤ n < N}, where N ∈ N ∪ {∞}, with values in a Borel space (A,A), and note Xn = (X1, . . . ,Xn). We say that X is Hoeffding decomposable if, for each n, every square integrable, centered and symmetric statistic based on Xn can be written as an orthogonal sum of n U statistics with degenerated and symmetric kernels of increasing order. The only two examples of Hoeffding decomposable sequences studied in the literature are i.i.d. random variables and extractions without replacement from a finite population. In the first part of the paper we establish a necessary and sufficient condition for an exchangeable sequence to be Hoeffding decomposable, that is, called weak independence. We show that not every exchangeable sequence is weakly independent, and, therefore, that not every exchangeable sequence is Hoeffding decomposable. In the second part we apply our results to a class of exchangeable and weakly independent random vectors