Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type $$C_n(B)=\sqrt{n}\{{\mu }_n(B)-P(X_{n+1}\in B\mid X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)\},$$ where $(X_n)$ is a sequence of random variables and ${\mu }_n=(1\mathrm{/}n){\sum }_{i=1}^n{\delta }_{X_i}$ the empirical measure. Conditions for ${\sup }_B\mathrm{\mid }{C}_n(B)\mathrm{\mid }$ 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 ${\sup }_B\mathrm{\mid }{C}_n(B)\mathrm{\mid }0$ or even that $\sqrt{n}{\sup }_B\mathrm{\mid }{C}_n(B)\mathrm{\mid }$ converges a.s. Results of this type are useful in Bayesian statistics.