Asymptotic behaviour of the empirical process for exchangeable data

Let SS be the space of real cadlag functions on RR with finite limits at ±∞±∞, equipped with uniform distance, and let XnXn be the empirical process for an exchangeable sequence of random variables. If regarded as a random element of SS, XnXn can fail to converge in distribution. However, in this paper, it is shown that E*f(Xn)→E*f(X)E*f(Xn)→E*f(X) for each bounded uniformly continuous function f   on SS, where X   is some (nonnecessarily measurable) random element of SS. 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 XnXn to converge in distribution are obtained.