eprintid: 980
rev_number: 7
eprint_status: archive
userid: 36
dir: disk0/00/00/09/80
datestamp: 2011-10-31 13:28:39
lastmod: 2011-11-03 13:19:36
status_changed: 2011-10-31 13:28:39
type: article
metadata_visibility: show
creators_name: Berti, Patrizia
creators_name: Crimaldi, Irene
creators_name: Pratelli, Luca
creators_name: Rigo, Pietro
creators_id: 
creators_id: irene.crimaldi@imtlucca.it
creators_id: 
creators_id: 
title: Rate of convergence of predictive distributions for dependent data
ispublished: pub
subjects: HA
subjects: QA
divisions: EIC
full_text_status: none
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)=\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.

date: 2009
publication: Bernoulli
volume: 15
number: 4
publisher: Bernoulli Society for Mathematical Statistics and Probability
pagerange: 1351-1367
id_number: 10.3150/09-BEJ191
refereed: TRUE
issn: 1350-7265
official_url: http://projecteuclid.org/euclid.bj/1262962239
citation:   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)