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

摘要

Given a recursive distributional equation (RDE) and a solution μ of it, we consider the tree indexed invariant process called the recursive tree process (RTP) with marginal μ. 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' construction of the frozen percolation process on a infinite regular tree (Aldous, 2000) and show that the associated RTP has a trivial tail.