Rate of convergence of predictive distributions for dependent data
Abstract
This paper deals with empirical processes of the type \[Cn(B)=n\μn(B)-P(Xn+1∈ B X1,...,Xn)\,\] where (Xn) is a sequence of random variables and μn=(1/n)Σi=1nδXi the empirical measure. Conditions for B|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 B|Cn(B)|P0 or even that nB|Cn(B)| converges a.s. Results of this type are useful in Bayesian statistics.
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