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Journal of Theoretical Probability

, Volume 31, Issue 2, pp 757–788 | Cite as

Critical Multi-type Galton–Watson Trees Conditioned to be Large

  • Romain AbrahamEmail author
  • Jean-François Delmas
  • Hongsong Guo
Article
  • 169 Downloads

Abstract

Under minimal condition, we prove the local convergence of a critical multi-type Galton–Watson tree conditioned on having a large total progeny by types toward a multi-type Kesten’s tree. We obtain the result by generalizing Neveu’s strong ratio limit theorem for aperiodic random walks on \(\mathbb {Z}^d\).

Keywords

Galton–Watson tree Random tree Local limit Strong ratio theorem Branching process 

Mathematics Subject Classification (2010)

60J80 60B10 

Notes

Acknowledgements

The authors would like to thank Jean-Philippe Chancelier for pointing out the references on convex analysis and his valuable advice as well as the two anonymous referees for their comments and suggestions. H. Guo would like to express her gratitude to J.-F. Delmas for his help during her stay at CERMICS. The research has also been supported by the ANR-14-CE25-0014 (ANR GRAAL).

References

  1. 1.
    Abraham, R., Delmas, J.-F.: Local limits of conditioned Galton–Watson trees: the condensation case. Electron. J. Probab. 19(56), 1–29 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abraham, R., Delmas, J.-F.: Local limits of conditioned Galton–Watson trees: the infinite spine case. Electron. J. Probab. 19(2), 1–19 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  4. 4.
    Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2006)zbMATHGoogle Scholar
  5. 5.
    Chaumont, L., Liu, R.: Coding multitype forests: application to the law of the total population of branching forests. Trans. Am. Math. Soc. 368, 2723–2747 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Delmas, J.-F., Hénard, O.: A Williams decomposition for spatially dependent superprocesses. Electron. J. Probab. 18(37), 1–43 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gnedenko, B.V.: On a local limit theorem of the theory of probability. Uspekhi Mat. Nauk 3(3), 187–194 (1948)MathSciNetGoogle Scholar
  8. 8.
    Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. English translation, Addison-Wesley, Cambridge (1954)Google Scholar
  9. 9.
    He, X.: Conditioning Galton–Watson trees on large maximal out-degree. J. Theor. Probab. (2016). doi: 10.1007/s10959-016-0664-x
  10. 10.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Janson, S.: Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9, 103–252 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jonnson, T., Stefansson, S.: Condensation in nongeneric trees. J. Stat. Phys. 142, 277–313 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kesten, H.: Subdiffusive behavior of random walk on a random cluster. Ann. de l’Inst. Henri Poincaré 22, 425–487 (1986)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kurtz, T., Lyons, R., Pemantle, R., Peres, Y.: A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes. In: Classical and modern branching processes (Minneapolis, 1994), volume 84 of IMA Vol. Math. Appl., pp. 181–185. Springer (1997)Google Scholar
  16. 16.
    Luis, J.A.L.-M., Gorostiza, G.: The multitype measure branching process. Adv. Appl. Probab. 22(1), 49–67 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miermont, G.: Invariance principles for spatial multitype Galton–Watson trees. Ann. Inst. H. Poincaré Probab. Statist 44, 1128–1161 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Neveu, J.: Sur le théorème ergodique de Chung-Erdős. C. R. Acad. Sci. Paris 257, 2953–2955 (1963)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pénisson, S.: Beyond Q-process: various ways of conditioning the multitype Galton–Watson process. ALEA 13, 223–237 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rizzolo, D.: Scaling limits of Markov branching trees and Galton–Watson trees conditioned on the number of vertices with out-degree in a given set. Ann. de l’Inst. Henri Poincaré 51(2), 512–532 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997)Google Scholar
  22. 22.
    Rvaceva, E.: On domains of attraction of multi-dimensional distributions. Sel. Transl. Math. Stat. Probab. 2, 183–205 (1961)MathSciNetGoogle Scholar
  23. 23.
    Spitzer, F.: Principles of Random Walk. Springer, Berlin (2013)zbMATHGoogle Scholar
  24. 24.
    Stephenson, R.: Local convergence of large critical multi-type Galton-Watson trees and applications to random maps. J. Theor. Probab. (2016). doi: 10.1007/s10959-016-0707-3
  25. 25.
    Stone, C.: On local and ratio limit theorems. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, no. (part II), pp. 217–224. University of California Press, Berkeley, Los Angeles (1966)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Romain Abraham
    • 1
    Email author
  • Jean-François Delmas
    • 2
  • Hongsong Guo
    • 2
    • 3
  1. 1.Laboratoire MAPMO, CNRS, UMR 7349, Fédération Denis Poisson, FR 2964Université d’OrléansOrléans Cedex 2France
  2. 2.Université Paris-Est, CERMICS (ENPC)Marne la ValléeFrance
  3. 3.Department of MathematicsChina University of Mining and TechnologyBeijingPeople’s Republic of China

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