Open Access
November 2009 Rate of convergence of predictive distributions for dependent data
Patrizia Berti, Irene Crimaldi, Luca Pratelli, Pietro Rigo
Bernoulli 15(4): 1351-1367 (November 2009). DOI: 10.3150/09-BEJ191


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.


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Patrizia Berti. Irene Crimaldi. Luca Pratelli. Pietro Rigo. "Rate of convergence of predictive distributions for dependent data." Bernoulli 15 (4) 1351 - 1367, November 2009.


Published: November 2009
First available in Project Euclid: 8 January 2010

zbMATH: 1375.60063
MathSciNet: MR2597596
Digital Object Identifier: 10.3150/09-BEJ191

Keywords: Bayesian predictive inference , central limit theorem , conditional identity in distribution , Empirical distribution , exchangeability , predictive distribution , stable convergence

Rights: Copyright © 2009 Bernoulli Society for Mathematical Statistics and Probability

Vol.15 • No. 4 • November 2009
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