Asymptotic behaviour of the empirical process for exchangeable data

Let S be the space of real cadlag functions on R with finite limits at ± ∞ , equipped with uniform distance, and let X n be the empirical process for an exchangeable sequence of random variables. If regarded as a random element of S , X n can fail to converge in distribution. However, in this paper, it is shown that E * f ( X n ) → E * f ( X ) 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 X n to converge in distribution are obtained.