A Necessary and Sufficient Condition for the Tail-Triviality of a Recursive Tree Process

Details A Necessary and Sufficient Condition for the Tail-Triviality of a Recursive Tree Process A Necessary and Sufficient Condition for the Tail-Triviality of a Recursive Tree Process

2005-11-08

arxiv.org/abs/math/0511203v2

Mathematics

Probability

Minor changes in the wordings. 20 pages, 1 figure. To appear in Sankhya

Scientific paper

Given a recursive distributional equation (RDE) and a solution $\mu$ of it, we consider the tree indexed invariant process called the recursive tree process (RTP) with marginal $\mu$. We introduce a new type of bivariate uniqueness property which is different from the one defined by Aldous and Bandyopadhyay (2005), and we prove that this property is equivalent to tail-triviality for the RTP, thus obtaining a necessary and sufficient condition to determine tail-triviality for a RTP in general. As an application we consider Aldous' (2000) construction of the frozen percolation process on a infinite regular tree and show that the associated RTP has a trivial tail.

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