## Exchangeable sequences driven by an absolutely continuous random measure.(English)Zbl 1277.60064

Summary: Let $$S$$ be a Polish space and $$(X_{n}:n\geq1)$$ an exchangeable sequence of $$S$$-valued random variables. Let $$\alpha_{n}(\cdot)=P(X_{n+1}\in\cdot\mid X_{1},\dotsc,X_{n})$$ be the predictive measure and $$\alpha$$ a random probability measure on $$S$$ such that $$\alpha_{n}\to\alpha$$ weakly 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 $$\mathcal{L}(X_{1},\dotsc,X_{n})$$, $$n\geq1$$, only. The other problem is to investigate whether $$\|\alpha_{n}-\alpha\|\to0$$ a.s., where $$\|\cdot\|$$ 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 [the authors, ibid. 32, No. 3A, 2029–2052 (2004; Zbl 1050.60004)].

### MSC:

 60G09 Exchangeability for stochastic processes 60G42 Martingales with discrete parameter 60G57 Random measures 62F15 Bayesian inference

Zbl 1050.60004
Full Text:

### References:

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