EXCHANGEABLE SEQUENCES DRIVEN BY AN ABSOLUTELY CONTINUOUS RANDOM MEASURE

• 成果类型: Article
• 署名作者: Berti, Patrizia; Pratelli, Luca; Rigo, Pietro
• 署名单位: Universita di Modena e Reggio Emilia; University of Pavia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP786
• 发表日期: 2013
• 页码: 2090-2102
• 关键词: markov moment problem; de-finettis theorem

摘要

Let S be a Polish space and (X-n : n >= 1) an exchangeable sequence of S-valued random variables. Let alpha(n)(.) = P(Xn+1 is an element of E . vertical bar X-1, ... X-n) be the preweak dictive measure and a a random probability measure on S such that an alpha(n) -> weak alpha a.s. Two (related) problems are addressed. One is to give conditions for alpha << lambda a.s., where lambda is a (nonrandom) sigma-finite Borel measure on S. Such conditions should concern the finite dimensional distributions L(X-1, ..., X-n), n >= 1, only. The other problem is to investigate whether parallel to alpha(n) - alpha parallel to -> a.s. 0, where parallel to . parallel to 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].