Exchangeable sequences driven by an absolutely continuous random measure

TL;DR

This paper investigates conditions under which exchangeable sequences driven by an absolutely continuous random measure have predictive measures absolutely continuous with respect to a fixed measure, and explores convergence properties of these measures.

Contribution

It provides new criteria for absolute continuity of predictive measures and analyzes convergence in total variation for exchangeable and conditionally identically distributed sequences.

Findings

Conditions for predictive measures to be absolutely continuous are established.

Convergence of predictive measures in total variation is demonstrated.

Results extend to conditionally identically distributed sequences beyond exchangeability.

Abstract

Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ 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 $\mathcal {L}(X_1,\...,X_n)$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alp ha_n-\alpha\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, 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].