The Annals of Probability

Automorphism Invariant Measures on Trees

Robin Pemantle

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Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176989706

Digital Object Identifier
doi:10.1214/aop/1176989706

Mathematical Reviews number (MathSciNet)
MR1175277

Zentralblatt MATH identifier
0760.05055

JSTOR
links.jstor.org

Subjects
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]

Keywords
Stationary tree exchangeable partially exchangeable mixing tail field

Citation

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


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