A new characterization of endogeny

Tibor Mach, Anja Sturm, Jan M. Swart

Published 2018-01-16Version 1

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 descendents. 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 answer this question positively. Our main tool is 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.