Rate of Convergence of Predictive Distributions for Dependent Data - Patrizia Berti; Irene Crimaldi; Luca Pratelli; Pietro Rigo (Università degli Studi di Pavia, Dipartimento di Economia Politica e Metodi Quantitativi (EPMQ), 2009)
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.
Keywords: 330; Bayesian predictive inference; Central limit theorem; Conditional identity in distribution; Empirical distribution; Exchangeability; Predictive distribution; Stable convergence