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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

Citations:

Zbl 1050.60004
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References:

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