A new characterization of endogeny

TL;DR

This paper introduces a higher-level recursive distributional equation to characterize endogeny in recursive tree processes, removing the need for continuity assumptions and providing a new criterion for endogeny.

Contribution

It develops a higher-level RDE framework that captures all multivariate RDEs and establishes a new criterion for endogeny based on fixed points of this RDE.

Findings

Higher-level RDE contains all n-variate RDEs via moment measures.

Minimal and maximal fixed points of the higher-level RDE coincide iff the RTP is endogenous.

The paper confirms that endogeny can be characterized without the continuity assumption.

Abstract

Aldous and Bandyopadhyay have shown that each solution to a recursive distributional equation (RDE) gives rise to recursive tree process (RTP), which is a sort of Markov chain in which time has a tree-like structure and in which the state of each vertex is a random function of its descendants. If the state at the root is measurable with respect to the sigma field generated by the random functions attached to all vertices, then the RTP is said to be endogenous. For RTPs defined by continuous maps, Aldous and Bandyopadhyay showed that endogeny is equivalent to bivariate uniqueness, and they asked if the continuity hypothesis can be removed. We introduce a higher-level RDE that through its $n$-th moment measures contains all $n$-variate RDEs. We show that this higher-level RDE has minimal and maximal fixed points with respect to the convex order, and that these coincide if and only if the corresponding RTP is endogenous. As a side result, this allows us to answer the question of Aldous and Bandyopadhyay positively.