Asymptotic behaviour of the empirical process for exchangeable data

Let be the space of real cadlag functions on with finite limits at ±[infinity], 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 , Xn can fail to converge in distribution. However, in this paper, it is shown that E*f(Xn)-->E*f(X) for each bounded uniformly continuous function f on , where X is some (nonnecessarily measurable) random element of . 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.