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Bernoulli

Rate of convergence of predictive distributions for dependent data

Patrizia Berti, Irene Crimaldi, Luca Pratelli, and Pietro Rigo
Source: Bernoulli Volume 15, Number 4 (2009), 1351-1367.

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.

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.bj/1262962239
Digital Object Identifier: doi:10.3150/09-BEJ191
Mathematical Reviews number (MathSciNet): MR2597596
Zentralblatt MATH identifier: 05816145

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