## The Annals of Applied Probability

### The lineage process in Galton–Watson trees and globally centered discrete snakes

Jean-François Marckert

#### Abstract

We consider branching random walks built on Galton–Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of “globally centered discrete snake” that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when n goes to +∞, “globally centered discrete snakes” converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton–Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node u is the vector indexed by (k, j) giving the number of ancestors of u having k children and for which u is a descendant of the jth one]. Some consequences concerning Galton–Watson trees conditioned by the size are also derived.

#### Article information

Source
Ann. Appl. Probab., Volume 18, Number 1 (2008), 209-244.

Dates
First available in Project Euclid: 9 January 2008

https://projecteuclid.org/euclid.aoap/1199890021

Digital Object Identifier
doi:10.1214/07-AAP450

Mathematical Reviews number (MathSciNet)
MR2380897

Zentralblatt MATH identifier
1140.60042

#### Citation

Marckert, Jean-François. The lineage process in Galton–Watson trees and globally centered discrete snakes. Ann. Appl. Probab. 18 (2008), no. 1, 209--244. doi:10.1214/07-AAP450. https://projecteuclid.org/euclid.aoap/1199890021

#### References

• Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis. (Durham, 1990). Lond. Math. Soc. Lect. Note Ser. 167 23–70. Cambridge Univ. Press.
• Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
• Aldous, D. (1993). Tree-based models for random distribution of mass. J. Statist. Phys. 73 625–641.
• Bousquet-Mélou, M. (2006). Limit laws for embedded trees: Applications to the integrated superBrownian excursion. Random Structures Algorithms 29 475–523.
• Bousquet-Mélou, M. and Janson, S. (2006). The density of the ISE and local limit laws for embedded trees. Ann. Appl. Probab. 16 1597–1632.
• Breuillard, E. (2005). Distributions diophantiennes et théorème limite local sur $\mathbbR^d$. Probab. Theory Related Fields 132 39–73.
• Chassaing, P. and Schaeffer, G. (2004). Random planar lattices and integrated superBrownian excursion. Probab. Theory Related Fields 128 161–212.
• Donsker, M. D. (1952). Justification and extension of Doob's heuristic approach to the Komogorov–Smirnov theorems. Ann. Math. Statist. 23 277–281.
• Duquesne, T. (2003). A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 996–1027.
• Duquesne, T. and Le Gall, J. F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281.
• Gittenberger, B. (2003). A note on “State spaces of the snake and its tour: Convergence of the discrete snake” by J.-F. Marckert and A. Mokkadem. J. Theoret. Probab. 16 1063–1067.
• Janson, S. (2005). Left and right pathlengths in random binary trees. Algorithmica 46 419–429.
• Janson, S. and Marckert, J.-F. (2005). Convergence of discrete snakes. J. Theoret. Probab. 18 615–647.
• Kallenberg, O. (1997). Foundations of Modern Probability. Probability and Its Applications. Springer, New York.
• Le Gall, J.-F. (2006). A conditional limit theorem for tree-indexed random walk. Stoch. Process. Appl. 116 539–567.
• Le Gall, J. F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
• Lyons, R. and Peres, Y. (2005). Probability on Trees and Networks. Available at http://php.indiana.edu/~rdlyons/prbtree/book.pdf.
• Marckert, J.-F. (2004). The rotation correspondence is asymptotically a dilatation. Random Structures Algorithms 24 118–132.
• Marckert, J.-F. and Miermont, G. (2007). Invariance principles for random bipartite planar maps. Ann. Probab. 35 1642–1705.
• Marckert, J.-F. and Mokkadem, A. (2003). States spaces of the snake and its tour: Convergence of the discrete snake. J. Theoret. Probab. 16 1015–1046.
• Marckert, J.-F. and Mokkadem, A. (2003). The depth first processes of Galton–Watson trees converge to the same Brownian excursion. Ann. Probab. 31 1655–1678.
• Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 199–207.
• Otter, R. (1949). The multiplicative process. Ann. Math. Statist. 20 206–224.
• Petrov, V. V. (1995). Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Univ. Press.
• Pitman, J. (2006). Combinatorial Stochastic Processes. Springer, Berlin.