Probability Theory and Related Fields

, Volume 159, Issue 3–4, pp 809–823 | Cite as

A hierarchical version of the de Finetti and Aldous-Hoover representations

Article
First Online:
Received:
Revised:

Abstract

We consider random arrays indexed by the leaves of an infinitary rooted tree of finite depth, with the distribution invariant under the rearrangements that preserve the tree structure. We call such arrays hierarchically exchangeable and prove that they satisfy an analogue of de Finetti’s theorem. We also prove a more general result for arrays indexed by several trees, which includes a hierarchical version of the Aldous-Hoover representation.

Keywords

Exchangeability Spin glasses 

Mathematics Subject Classification (2010)

60G09 60K35 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We would like to thank the referees for their careful review and a number of suggestions to improve the quality of the paper.

References

  1. 1.
    Aldous, D.: Representations for partially exchangeable arrays of random variables. J. Multivar. Anal. 11(4), 581–598 (1981)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aldous, D.: Exchangeability and related topics. In: École d’été probabilités de Saint-Flour, XIII-1983, Lecture Notes in Math., 1117, pp. 1–198. Springer, Berlin (1985)Google Scholar
  3. 3.
    Aldous, D.: More uses of exchangeability: representations of complex random structures. In: Probability and Mathematical Genetics, London Mathematics Society. Lecture Note Series 378, pp. 35–63. Cambridge Univ. Press, Cambridge (2010)Google Scholar
  4. 4.
    Austin, T.: On exchangeable random variables and the statistics of large graphs and hypergraphs. Probab. Surv. 5, 80–145 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Austin, T.: Exchangeable random measures. Preprint, arXiv:1302.2116 (2013)Google Scholar
  6. 6.
    Dudley, R. M.: Real analysis and probability. In: Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge (2002)Google Scholar
  7. 7.
    Glasner, E.: Ergodic theory via joinings. In: Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence (2003)Google Scholar
  8. 8.
    Hoover, D.N.: Relations on probability spaces. Preprint (1979)Google Scholar
  9. 9.
    Hoover, D.N.: Row-column exchangeability and a generalized model for probability. In: Exchangeability in Probability and Statistics (Rome, 1981), pp. 281–291. Amsterdam, North-Holland (1982)Google Scholar
  10. 10.
    Kallenberg, O.: On the representation theorem for exchangeable arrays. J. Multivar. Anal. 30(1), 137–154 (1989)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kallenberg, O.: Probabilistic Symmetries and Invariance Principles. Probab. Appl. Springer-Verlag, New York (2005)Google Scholar
  12. 12.
    Mézard, M., Parisi, G.: The Bethe lattice spin glass revisited. Eur. Phys. J. B Condens. Matter Phys. 20(2), 217–233 (2001)MathSciNetGoogle Scholar
  13. 13.
    Panchenko, D.: The Parisi ultrametricity conjecture. Ann. Math. (2) 177(1), 383–393 (2013)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Panchenko, D.: Hierarchical exchangeability of pure states in mean field spin glass models. Preprint, arXiv:1307.2207 (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations