Rate of Convergence of Predictive Distributions for Dependent Data

Abstract

This paper deals with empirical processes of the type Cn(B) = n^(1/2) {µn(B) - P(Xn+1 in B | X1, . . . ,Xn)} , where (Xn) is a sequence of random variables and µn = (1/n)SUM(i=1,..,n) d(Xi) 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 [6]). By such conditions, in some relevant situations, one obtains that supB|Cn(B)|-P->0 or even that n^(1/2) supB|Cn(B)| converges a.s.. Results of this type are useful in Bayesian statistics.Bayesian predictive inference, Central limit theorem, Conditional identity in distribution, Empirical distribution, Exchangeability, Predictive distribution, Stable convergence.