The Annals of Probability

Automorphism Invariant Measures on Trees

Robin Pemantle

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Consider a collection of real-valued random variables indexed by 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.

Article information

Ann. Probab., Volume 20, Number 3 (1992), 1549-1566.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]
Secondary: 28D99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Stationary tree exchangeable partially exchangeable mixing tail field


Pemantle, Robin. Automorphism Invariant Measures on Trees. Ann. Probab. 20 (1992), no. 3, 1549--1566. doi:10.1214/aop/1176989706.

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