A Central Limit Theorem for Predictive Distributions

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