Bernoulli

  • Bernoulli
  • Volume 15, Number 4 (2009), 1351-1367.

Rate of convergence of predictive distributions for dependent data

Patrizia Berti, Irene Crimaldi, Luca Pratelli, and Pietro Rigo

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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.

Article information

Source
Bernoulli, Volume 15, Number 4 (2009), 1351-1367.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1262962239

Digital Object Identifier
doi:10.3150/09-BEJ191

Mathematical Reviews number (MathSciNet)
MR2597596

Zentralblatt MATH identifier
1375.60063

Keywords
Bayesian predictive inference central limit theorem conditional identity in distribution empirical distribution exchangeability predictive distribution stable convergence

Citation

Berti, Patrizia; Crimaldi, Irene; Pratelli, Luca; Rigo, Pietro. Rate of convergence of predictive distributions for dependent data. Bernoulli 15 (2009), no. 4, 1351--1367. doi:10.3150/09-BEJ191. https://projecteuclid.org/euclid.bj/1262962239


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