Asymptotics of predictive distributions

Abstract

Let (Xn) be a sequence of random variables, adapted to a filtration (Gn), and let μn = (1/n) ∑ni=1 δXi and an(・) = P(Xn+1∈ ・ | Gn) be the empirical and the predictive measures. We focus on ||μn − an|| = supB∈D |μn(B) − an(B)|, where D is a class of measurable sets. Conditions for ||μn − an|| → 0, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of rn ||μn − an|| is investigated for suitable constants rn. Special attention is paid to rn = √n. The sequence (Xn) is exchangeable or, more generally, conditionally identically distributed.