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 n - a(a.s.) --> 0, whereis 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 .
Conditional identity in distribution Exchangeability Predictive measure Random probability measure