Exchangeable sequences driven by an absolutely continuous random measure

Abstract

Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $α_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ be the predictive measure and $α$ a random probability measure on $S$ such that $α_n\stackrel{\mathrm{weak}}{\longrightarrow}α$ a.s. Two (related) problems are addressed. One is to give conditions for $α\llλ$ a.s., where $λ$ is a (nonrandom) $σ$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal {L}(X_1,\...,X_n)$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alp ha_n-α\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert\cdot\Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_n)$ is conditionally identically distributed, in the sense of [Ann. Probab. 32 (2004) 2029-2052].