A central limit theorem for predictive distributions

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(⋅)=P(Xn+1∈⋅∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan−→−a.s.∫fdμ for all f∈F. Define Dn(f)=dn{∫fdan−∫fdμ} for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn={Dn(f):f∈F} stably converge to a Gaussian kernel with a known covariance structure. In addition, E{φ(Dn(f))∣X1,…,Xn} converges in probability for all f∈F and φ∈Cb(R).