Sankhya: The Indian Journal of Statistics

2006, Volume 68, Pt. 1, 1--23

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

Antar Bandyopadhyay, Chalmers University of Technology, Sweden

SUMMARY. Given a \emph{recursive distributional equation} (RDE) and a solution $\mu$ of it, we consider the tree indexed invariant process called the \emph{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' construction of the frozen percolation process on a infinite regular tree (Aldous, 2000) and show that the associated RTP has a trivial tail.

AMS (1991) subject classification. Primary 60K35, 60G10, 60G20.

Key words and phrases. Bivariate uniqueness, distributional identities, endogeny, fixed point equations, frozen percolation process, recursive distributional equations, recursive tree process, tail-triviality.

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