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

Abstract

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.

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

References in corpus (1)

• A survey of max-type recursive distributional equations

Cited by in corpus (2)

• A new characterization of endogeny
• Bivariate Uniqueness and Endogeny for the Logistic Recursive Distributional Equation