@article{eprints980,
          number = {4},
          volume = {15},
          author = {Patrizia Berti and Irene Crimaldi and Luca Pratelli and Pietro Rigo},
            year = {2009},
         journal = {Bernoulli},
           title = {Rate of convergence of predictive distributions for dependent data},
           pages = {1351--1367},
       publisher = {Bernoulli Society for Mathematical Statistics and Probability},
             url = {http://eprints.imtlucca.it/980/},
        keywords = {Bayesian predictive inference; central limit theorem; conditional identity in distribution; empirical distribution; exchangeability; predictive distribution; stable convergence},
        abstract = {This paper deals with empirical processes of the type

[C\_\{n\}(B)={$\backslash$}sqrt\{n\}{$\backslash$}\{{$\backslash$}mu\_\{n\}(B)-P(X\_\{n+1\}{$\backslash$}in B{$\backslash$}mid X\_\{1\},{$\backslash$}ldots,X\_\{n\}){$\backslash$}\},{$\backslash$}]

where (Xn) is a sequence of random variables and {\ensuremath{\mu}}n=(1/n)?i=1n{\ensuremath{\delta}}Xi the empirical measure. Conditions for supB{\ensuremath{|}}Cn(B){\ensuremath{|}} 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 \${$\backslash$}sup\_\{B\}{\ensuremath{|}}C\_\{n\}(B){\ensuremath{|}}{$\backslash$}stackrel\{P\}\{{$\backslash$}rightarrow\}0\$ or even that \${$\backslash$}sqrt\{n\}{$\backslash$}sup\_\{B\}{\ensuremath{|}}C\_\{n\}(B){\ensuremath{|}}\$ converges a.s. Results of this type are useful in Bayesian statistics.
}
}