A New Characterization of Endogeny

  • Tibor Mach
  • Anja Sturm
  • Jan M. SwartEmail author


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.


Recursive tree process Endogeny 

Mathematics Subject Classification (2010)

60K35 60J05 82C22 60J80 



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.


  1. 1.
    Aldous, D.J., Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15(2), 1047–1110 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bandyopadhyay, A.: A necessary and sufficient condition for the tail-triviality of a recursive tree process. Sankhya 68(1), 1–23 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brossard, J., Leuridan, C.: Chaînes de Markov consructives indexées par ℤ. Ann. Probab. 35(2), 715–731 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourbaki, N.: Éléments De Mathématique. VIII. Part. 1: Les Structures Fondamentales De l’Analyse. Livre III: Topologie Générale. Chap. 9: Utilisation Des Nombres Réels en Topologie Générale. 2iéme éd. Actualités Scientifiques Et Industrielles 1045. Hermann & Cie, Paris (1958)zbMATHGoogle Scholar
  5. 5.
    Carothers, N.L.: Real analysis. Cambridge University Press (2000)Google Scholar
  6. 6.
    Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Springer, Berlin (1977)CrossRefGoogle Scholar
  7. 7.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions, 2nd. De Gruyter, Berlin (2011)CrossRefGoogle Scholar
  8. 8.
    Mossel, E.: Reconstruction on trees: beating the second eigenvalue. Ann. Appl. Probab. 11(1), 285–300 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Struct. Algoritm. 9, 223–252 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Rosenblatt, M.: Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8, 665–681 (1959)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439 (1965)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.The Czech Academy of SciencesInstitute of Information Theory and AutomationPrague 8Czech Republic
  2. 2.Institute for Mathematical StochasticsGeorg-August-Universität GöttingenGöttingenGermany

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