Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations

Abstract

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 Ustatistics 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 X (α,c) n = (X (α,c) 1 , . . . , X (α,c) n ) whose law is characterized by a positive and finite measure α(•) on A and by a real constant c. For instance, if c = 0, X (α,c) n

Hoeffding spaces and Specht modules

“…A random variable such as (4.1) is called a U -statistic based on X (m) , with a symmetric kernel φ of order l. One has that SU l ⊂ SU l+1 (see e.g. [9]) and SU m = L 2 s X (m) . The collection of the symmetric Hoeffding spaces associated to X (m) , noted {SH l : l = 0, ..., m} is defined as follows: SH 0 = SU 0 , and…”

“…The following result can be proved by using the content of [3], Section 2, or as a special case of [9], Theorem 11. …”

“…Completely degenerated kernels are related to the notion of weak independence in [9], Theorem 6. Note that, in the above quoted references, the content of Proposition 4.1 is proved for real valued symmetric statistics (the extension of such results to complex random variables is immediate: just consider separately the real and the imaginary parts of each statistic).…”

“…See e.g. [12], or the references indicated in the introduction to [9], for further discussions in this direction.…”

“…See El-Dakkak and Peccati [7] and Peccati [9] for some general statements; see Bloznelis [2], Bloznelis and Götze [3,4] and Zhao and Chen [14] for a comprehensive analysis of Hoeffding decompositions associated with extractions without replacement from a finite population.…”