%0 Journal Article
%@ 1350-7265
%A Berti, Patrizia
%A Crimaldi, Irene
%A Pratelli, Luca
%A Rigo, Pietro
%D 2009
%F eprints:980
%I Bernoulli Society for Mathematical Statistics and Probability
%J Bernoulli
%K Bayesian predictive inference; central limit theorem; conditional identity in distribution; empirical distribution; exchangeability; predictive distribution; stable convergence
%N 4
%P 1351-1367
%T Rate of convergence of predictive distributions for dependent data
%U http://eprints.imtlucca.it/980/
%V 15
%X 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 (Xn) is a sequence of random variables and μn=(1/n)∑i=1nδXi the empirical measure. Conditions for supB|Cn(B)| to converge stably (in particular, in distribution) are given, where B ranges over a suitable class of measurable sets. These conditions apply when (Xn) 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.