Invariance principles for spatial multitype Galton–Watson trees

Grégory Miermont

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 1128-1161
  • ISSN: 0246-0203

Abstract

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We prove that critical multitype Galton–Watson trees converge after rescaling to the brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the brownian snake, under some moment assumptions.

How to cite

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Miermont, Grégory. "Invariance principles for spatial multitype Galton–Watson trees." Annales de l'I.H.P. Probabilités et statistiques 44.6 (2008): 1128-1161. <http://eudml.org/doc/78006>.

@article{Miermont2008,
abstract = {We prove that critical multitype Galton–Watson trees converge after rescaling to the brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the brownian snake, under some moment assumptions.},
author = {Miermont, Grégory},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {multitype Galton–Watson tree; discrete snake; invariance principle; brownian tree; brownian snake; multitype Galton-Watson tree; Brownian tree; Brownian snake},
language = {eng},
number = {6},
pages = {1128-1161},
publisher = {Gauthier-Villars},
title = {Invariance principles for spatial multitype Galton–Watson trees},
url = {http://eudml.org/doc/78006},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Miermont, Grégory
TI - Invariance principles for spatial multitype Galton–Watson trees
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 1128
EP - 1161
AB - We prove that critical multitype Galton–Watson trees converge after rescaling to the brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the brownian snake, under some moment assumptions.
LA - eng
KW - multitype Galton–Watson tree; discrete snake; invariance principle; brownian tree; brownian snake; multitype Galton-Watson tree; Brownian tree; Brownian snake
UR - http://eudml.org/doc/78006
ER -

References

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  1. [1] D. J. Aldous. The continuum random tree. III. Ann. Probab. 21 (1993) 248–289. Zbl0791.60009MR1207226
  2. [2] K. B. Athreya and P. E. Ney. Branching Processes. Springer, New York, 1972. (Die Grundlehren der mathematischen Wissenschaften, Band 196.) Zbl0259.60002MR373040
  3. [3] T. Duquesne. A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 (2003) 996–1027. Zbl1025.60017MR1964956
  4. [4] T. Duquesne and J.-F. Le Gall. Random trees, Lévy processes and spatial branching processes. Astérisque 281 (2002) vi + 147. Zbl1037.60074MR1954248
  5. [5] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553–603. Zbl1070.60076MR2147221
  6. [6] W. Feller. An Introduction to Probability Theory and Its Applications Vol. II, 2nd edition. Wiley, New York, 1971. Zbl0219.60003MR270403
  7. [7] T. E. Harris. The Theory of Branching Processes. Dover Phoenix Editions. Dover Publications Inc., Mineola, NY, 2002. (Corrected reprint of the 1963 original. Springer, Berlin) Zbl1037.60001MR163361
  8. [8] P. Jagers. General branching processes as Markov fields. Stochastic Process. Appl. 32 (1989) 183–212. Zbl0678.92009MR1014449
  9. [9] S. Janson. Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 (2005) 417–452. Zbl1112.60012MR2226887
  10. [10] S. Janson and J.-F. Marckert. Convergence of discrete snakes. J. Theoret. Probab. 18 (2005) 615–645. Zbl1084.60049MR2167644
  11. [11] T. Kurtz, R. Lyons, R. Pemantle and Y. Peres. A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994) 181–185. IMA Vol. Math. Appl. 84. Springer, New York, 1997. Zbl0868.60068MR1601737
  12. [12] J.-F. Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel, 1999. Zbl0938.60003MR1714707
  13. [13] J.-F. Marckert and G. Miermont. Invariance principles for random bipartite planar maps. Ann. Probab. 35 (2007) 1642–1705. Zbl1208.05135MR2349571
  14. [14] G. Miermont. An invariance principle for random planar maps. In Fourth Colloquium on Mathematics and Computer Sciences CMCS’06, Discrete Math. Theor. Comput. Sci. Proc., AG 39–58 (electronic). Nancy, 2006. Zbl1195.60049MR2509622
  15. [15] V. V. Petrov. Limit Theorems of Probability Theory. The Clarendon Press Oxford University Press, New York, 1995. Zbl0826.60001MR1353441
  16. [16] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer, Berlin, 1999. Zbl0917.60006MR1725357
  17. [17] E. Seneta. Nonnegative Matrices and Markov Chains, 2nd edition. Springer, New York, 1981. Zbl0471.60001MR719544
  18. [18] D. W. Stroock. Probability Theory, an Analytic View. Cambridge Univ. Press, 1993. Zbl0925.60004MR1267569
  19. [19] V. A. Vatutin and E. E. Dyakonova. The survival probability of a critical multitype Galton–Watson branching process. J. Math. Sci. (New York) 106 (2001) 2752–2759. Zbl0999.60081MR1878742
  20. [20] L. M. Wu. Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23 (1995) 420–445. Zbl0828.60017MR1330777

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