Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Local limits of large Galton–Watson trees rerooted at a random vertex

Benedikt Stufler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton–Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.

Résumé

Nous discutons de plusieurs formes de convergence du voisinage d’un sommet aléatoire uniforme dans des arbres aléatoires simplement générés, lorsque leur taille tend vers l’infini. Pour le cas standard d’un arbre de Galton–Watson critique conditionné à être grand, la limite est le sin-tree invariant aléatoire construit par Aldous (1991). Dans le régime de condensation, nous décrivons en toute généralité le comportement asymptotique local depuis un sommet aléatoire jusqu’à son premier ancêtre de grand degré. Au delà de cet ancêtre distingué, différents comportements peuvent apparaître selon les poids de branchement. Dans un sous-régime de condensation complète, nous obtenons la convergence vers un nouvel arbre limite, qui décrit la forme asymptotique du voisinage du chemin complet depuis un sommet aléatoire jusqu’à la racine. Cela inclut le cas où la distribution de la descendance suit une loi de puissance, à un facteur près qui varie lentement à l’infini.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 155-183.

Dates
Received: 5 November 2016
Revised: 19 December 2017
Accepted: 21 December 2017
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802398

Digital Object Identifier
doi:10.1214/17-AIHP879

Mathematical Reviews number (MathSciNet)
MR3901644

Zentralblatt MATH identifier
07039768

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures

Keywords
Local weak limits Simply generated trees Fringe distributions

Citation

Stufler, Benedikt. Local limits of large Galton–Watson trees rerooted at a random vertex. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 155--183. doi:10.1214/17-AIHP879. https://projecteuclid.org/euclid.aihp/1547802398


Export citation

References

  • [1] R. Abraham, A. Bouaziz and J.-F. Delmas. Local limits of Galton–Watson trees conditioned on the number of protected nodes. J. Appl. Probab. 54 (1) (2017) 55–65.
  • [2] R. Abraham and J.-F. Delmas. Local limits of conditioned Galton–Watson trees: The condensation case. Electron. J. Probab. 19 (2014) Article ID 56.
  • [3] R. Abraham and J.-F. Delmas. Local limits of conditioned Galton–Watson trees: The infinite spine case. Electron. J. Probab. 19 (2014) Article ID 2.
  • [4] R. Abraham, J.-F. Delmas and H. Guo. Critical multi-type Galton–Watson trees conditioned to be large. ArXiv e-print, 2015. Available at arXiv:1511.01721.
  • [5] D. Aldous. Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 (2) (1991) 228–266.
  • [6] I. Armendáriz and M. Loulakis. Conditional distribution of heavy tailed random variables on large deviations of their sum. Stochastic Process. Appl. 121 (5) (2011) 1138–1147.
  • [7] P. Billingsley. Weak Convergence of Measures: Applications in Probability. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 5. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1971.
  • [8] L. Devroye and S. Janson. Distances between pairs of vertices and vertical profile in conditioned Galton–Watson trees. Random Structures Algorithms 38 (4) (2011) 381–395.
  • [9] M. Drmota. Random Trees. An Interplay Between Combinatorics and Probability. Springer, New York, 2009.
  • [10] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010.
  • [11] C. Holmgren and S. Janson. Fringe trees, Crump–Mode–Jagers branching processes and $m$-ary search trees. Probab. Surv. 14 (2017) 53–154.
  • [12] S. Janson. Random cutting and records in deterministic and random trees. Random Structures Algorithms 29 (2) (2006) 139–179.
  • [13] S. Janson. Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9 (2012) 103–252.
  • [14] S. Janson, T. Jonsson and S. Ö. Stefánsson. Random trees with superexponential branching weights. J. Phys. A 44(48) (2012) Article ID 485002.
  • [15] T. Jonsson and S. Ö. Stefánsson. Condensation in nongeneric trees. J. Stat. Phys. 142 (2) (2011) 277–313.
  • [16] I. Kortchemski. Limit theorems for conditioned non-generic Galton–Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 51 (2) (2015) 489–511.
  • [17] S. Pénisson. Beyond the $Q$-process: Various ways of conditioning the multitype Galton–Watson process. ALEA Lat. Am. J. Probab. Math. Stat. 13 (1) (2016) 223–237.
  • [18] R. Stephenson. Local convergence of large critical multi-type Galton–Watson trees and applications to random maps. J. Theoret. Probab. 29 (2016) 1–47. 10.1007/s10959-016-0707-3.