Hoeffding decompositions and urn sequences

Abstract

Let X=(X₁, X₂, …) be a nondeterministic infinite exchangeable sequence with values in {0, 1}. We show that X is Hoeffding decomposable if, and only if, X is either an i.i.d. sequence or a Pólya sequence. This completes the results established in Peccati [Ann. Probab. 32 (2004) 1796–1829]. The proof uses several combinatorial implications of the correspondence between Hoeffding decomposability and weak independence. Our results must be compared with previous characterizations of i.i.d. and Pólya sequences given by Hill, Lane and Sudderth [Ann. Probab. 15 (1987) 1586–1592] and Diaconis and Ylvisaker [Ann. Statist. 7 (1979) 269–281]. The final section contains a partial characterization of Hoeffding decomposable sequences with values in a set with more than two elements.