Rate of convergence of predictive distributions for dependent data

Abstract

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{μ_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and ${\mu }_n=(1/n){\sum }_{i=1}^n{\delta }_{X_i}$ 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 ${\sup }_B|C_n(B)|\stackrel{P}{\to }0$ or even that $\sqrt{n}{\sup }_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

Published in at http://dx.doi.org/10.3150/09-BEJ191 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Limit theorems for a class of identically distributed random variables
• Rate of convergence of predictive distributions for dependent data
• Forward estimation for ergodic time series

Cited by in corpus (2)

• Rate of convergence of predictive distributions for dependent data
• Central limit theorems for an Indian buffet model with random weights