Exchangeable Sequences Driven by an Absolutely Continuous Random Measure

Let S be a Polish space and (Xn : n = 1) an exchangeable sequence of S-valued random variables. Let an(·) = P( Xn+1 in · | X1, . . . ,Xn) be the predictive measure and a a random probability measure on S such that an (weak) --> a a.s.. Two (related) problems are addressed. One is to give conditions for a << l a.s., where l is a (non random) sigma-finite Borel measure on S. Such conditions should concern the finite dimensional distributions L(X1, . . . ,Xn), n = 1, only. The other problem is to investigate whether ||an - a|| (a.s.) --> 0, where ||·|| is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that (Xn) is conditionally identically distributed, in the sense of [2].