Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type [C_{n}(B)=\sqrt{n}\{\mu_{n}(B)-P(X_{n+1}\in B|X_{1},\ldots,X_{n})\},\] where (X_n) is a sequence of random variables and μ_n=(1/n)∑_i=1^n\delta_Xi the empirical measure. Conditions for sup_B|C_n(B)| to converge stably (in particular, in distribution) are given, where B ranges over a suitable class of measurable sets. These conditions apply when (X_n) is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029–2052]). By such conditions, in some relevant situations, one obtains that $\sup_{B}|C_{n}(B)|\stackrel{P}{\rightarrow}0$ or even that $\sqrt{n}\sup_{B}|C_{n}(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.