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 & RARR;R. Let an(& BULL;)=P(Xn+1 & ISIN;& BULL; divide X1, horizontal ellipsis ,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 mu on S such that integral fdan ? a.s.integral fd mu for all f & ISIN;F. Define Dn(f)=dn integral fdan-integral fd mu for all f & ISIN;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 & ISIN;F stably converge to a Gaussian kernel with a known covariance structure. In addition, E phi(Dn(f)) divide X1, horizontal ellipsis ,Xn converges in probability for all f & ISIN;F and phi & ISIN;Cb(R).