AUTOMORPHISM INVARIANT-MEASURES ON TREES

• 成果类型: Article
• 署名作者: PEMANTLE, R
• 署名单位: Cornell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989706
• 发表日期: 1992
• 页码: 1549-1566
• 关键词: random-fields

摘要

Consider a collection of real-valued random variables indexed the integers. It is well known that such a process can be stationary, that is, translation invariant, and ergodic and yet have very strong associations: The one-sided tail field may determine the sample; the measure may fail to be mixing in any sense; the weak law of large numbers may fail on some infinite subset of the integers. The main result of this paper is that this cannot happen if the integers are replaced by an infinite homogeneous tree and the translations are replaced by all graph automorphisms. In fact, any automorphism-invariant process indexed by the tree is a mixture of extremal processes whose one-sided tail fields are trivial, from which the mixing properties follow.