Asymptotic behaviour of the empirical process for exchangeable data

Let S be the space of real cadlag functions on R with finite limits at 1, equipped with uniform distance, and let Xn be the empirical process for an exchangeable sequence of random variables. If regarded as a random element of S, Xn can fail to converge in distribution. However, in this paper, it is shown that Ef ðXnÞ ! Ef ðXÞ for each bounded uniformly continuous function f on S, where X is some (nonnecessarily measurable) random element of S. In view of this fact, among other things, a conjecture raised in [P. Berti, P. Rigo, Convergence in distribution of nonmeasurable random elements, Ann. Probab. 32 (2004) 365–379] is settled and necessary and sufficient conditions for Xn to converge in distribution are obtained.