The contour of splitting trees is a lévy process

Annals of Probability

Volumn 38, Issue 1, 2010, Pages 348-395

  Lambert, Amaury ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Coalescent point process; Composition of subordinators; Contour process; Crump Mode Jagers branching process; Exploration process; Jirina process; L vy process; limit theorems; Malthusian parameter; Modified geometric distribution.; Poisson point process; Population dynamics; Real trees; Scale function; Yaglom distribution

Indexed keywords

EID: 77953681212     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOP485     Document Type: Article

References (26)

1. Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1-28.
2. Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248-289.
3. Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
4. Bertoin, J., Fontbona, J. and Martínez, S. (2008). On prolific individuals in a supercritical continuous-state branching process. J. Appl. Probab. 45 714-726.
5. Bertoin, J. and Le Gall, J.-F. (2000). The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 249-266.
6. Dress, A. W. M. and Terhalle, W. F. (1996). The real tree. Adv. Math. 120 283-301.
7. Duquesne, T. (2007). The coding of compact real trees by real valued functions. Preprint. Available at arXiv
8. Duquesne, T. and Le Gall, J. F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 1-147.
9. Evans, S. N. (2009). Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin.
10. Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Math. Appl. 84 111-126. Springer, New York.
11. Jaffard, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207-227.
12. Jiřina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8 292-313.
13. Lambert, A. (2002). The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 42-70.
14. Lambert, A. (2003). Coalescence times for the branching process. Adv. in Appl. Probab. 35 1071-1089.
15. Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24 45-163.
16. Lambert, A. (2009). The allelic partition for coalescent point processes. Markov Processes Relat. Fields. Preprint. To appear. Available at arXiv
17. Lambert, A. (2010). Spine decompositions of Lévy trees. To appear. In preparation.
18. Le Gall, J.-F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-383.
19. Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245-311 (electronic).
20. Le Gall, J.-F. and LE JAN, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213-252.
21. Neveu, J. (1986). Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 199-207.
22. Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57 365-395.
23. O'connell, N. (1995). The genealogy of branching processes and the age of our most recent common ancestor. Adv. in Appl. Probab. 27 418-442.
24. Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14 2120-2148.
25. Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
26. TAÏB, Z. (1992). Branching Processes and Neutral Evolution. Lecture Notes in Biomathematics 93. Springer, Berlin.