# A New Characterization of Endogeny

## Abstract

Aldous and Bandyopadhyay have shown that each solution to a recursive distributional equation (RDE) gives rise to a 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.

## Keywords

Recursive tree process Endogeny## Mathematics Subject Classification (2010)

60K35 60J05 82C22 60J80## Notes

### Acknowledgements

We thank Wolfgang Löhr and Jan Seidler for their help with Lemmas 11 and 12, respectively. We thank David Aldous, Antar Bandyopadhyay, and Christophe Leuridan for answering our questions about their work.

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