Rate of convergence of predictive distributions for dependent data

Berti, Patrizia and Crimaldi, Irene and Pratelli, Luca and Rigo, Pietro Rate of convergence of predictive distributions for dependent data. Bernoulli, 15 (4). pp. 1351-1367. ISSN 1350-7265 (2009)

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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 (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.

Item Type: Article 10.3150/09-BEJ191 Bayesian predictive inference; central limit theorem; conditional identity in distribution; empirical distribution; exchangeability; predictive distribution; stable convergence H Social Sciences > HA StatisticsQ Science > QA Mathematics Economics and Institutional Change Irene Crimaldi 31 Oct 2011 13:28 03 Nov 2011 13:19 http://eprints.imtlucca.it/id/eprint/980