Rate of convergence of predictive distributions for dependent data

Abstract

This paper deals with empirical processes of the type $$C_{n}(B)=\sqrt{n}\{\mu_{n}(B)-P(X_{n+1}\in B\mid X_{1},\ldots,X_{n})\},$$ where $(X_n)$ is a sequence of random variables and $μ_n=(1/n)∑_{i=1}^nδ_{X_i}$ 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.