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 $\alpha_{n}(\cdot)=P(X_{n+1}\in\cdot\mid X_{1},\ldots,X_{n})$ be the predictive measure and $\alpha$ a random probability measure on $S$ such that $\alpha_{n}\stackrel{\mathrm{weak}}{\longrightarrow}\alpha$ a.s. Two (related) problems are addressed. One is to give conditions for $\alpha\ll\lambda$ a.s., where $\lambda$ is a (nonrandom) $\sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal{L}(X_{1},\ldots,X_{n})$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alpha_{n}-\alpha\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, 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].