The coalescent point process of branching trees

Annals of Applied Probability

Volumn 23, Issue 1, 2013, Pages 99-144

  Lambert, Amaury ^a   Popovic, Lea ^a  

^a not available

Author keywords

Branching process; Coalescent point process; Continuous state branching process; Excursion; Feller diffusion; Height process; Linear fractional distribution; Multiple ancestry; Poisson point process; Quasi stationary distribution

Indexed keywords

EID: 84879682945     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AAP820     Document Type: Article

References (28)

1. ALDOUS, D. and POPOVIC, L. (2005). A critical branching process model for biodiversity. Adv. in Appl. Probab. 37 1094-1115. MR2193998
2. ATHREYA, K. B. and NEY, P. E. (1972). Branching Processes. Die Grundlehren der Mathematischen Wissenschaften, Band 196. Springer, New York. MR0373040
3. BENNIES, J. and KERSTING, G. (2000). A random walk approach to Galton-Watson trees. J. Theoret. Probab. 13 777-803. MR1785529
4. 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. MR1771663
5. BERTOIN, J. and LE GALL, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147-181 (electronic). MR2247827
6. BIRKNER, M., BLATH, J., CAPALDO, M., ETHERIDGE, A., MÖHLE, M., SCHWEINSBERG, J. andWAKOLBINGER, A. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 303-325 (electronic). MR2120246
7. CHAUVIN, B., ROUAULT, A. and WAKOLBINGER, A. (1991). Growing conditioned trees. Stochastic Process. Appl. 39 117-130. MR1135089
8. DUQUESNE, T. and LE GALL, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147. MR1954248
9. EVANS, S. N. (1993). Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959-971. MR1249698
10. FLEISCHMANN, K. and SIEGMUND-SCHULTZE, R. (1977). The structure of reduced critical Galton-Watson processes. Math. Nachr. 79 233-241. MR0461689
11. GEIGER, J. (1996). Size-biased and conditioned random splitting trees. Stochastic Process. Appl. 65 187-207. MR1425355
12. GEIGER, J. (1999). Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab. 36 301-309. MR1724856
13. GERNHARD, T. (2008). The conditioned reconstructed process. J. Theoret. Biol. 253 769-778.
14. GRIMVALL, A. (1974). On the convergence of sequences of branching processes. Ann. Probab. 2 1027-1045. MR0362529
15. KALLENBERG, O. (1977). Stability of critical cluster fields. Math. Nachr. 77 7-43. MR0443078
16. KALLENBERG, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York. MR1876169
17. KINGMAN, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235-248. MR0671034
18. LAMBERT, A. (2002). The genealogy of continuous-state branching processes with immigration. Probab. Theory Related Fields 122 42-70. MR1883717
19. LAMBERT, A. (2003). Coalescence times for the branching process. Adv. in Appl. Probab. 35 1071-1089. MR2014270
20. LAMBERT, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 420-446. MR2299923
21. LAMBERT, A. (2008). Population dynamics and random genealogies. Stoch. Models 24 45-163. MR2466449
22. LAMBERT, A. (2009). The allelic partition for coalescent point processes. Markov Process. Related Fields 15 359-386. MR2554367
23. LAMBERT, A. (2010). The contour of splitting trees is a Lévy process. Ann. Probab. 38 348-395. MR2599603
24. LE GALL, J.-F. and LE JAN, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213-252. MR1617047
25. LYONS, R., PEMANTLE, R. and PERES, Y. (1995). Conceptual proofs of LlogL criteria for mean behavior of branching processes. Ann. Probab. 23 1125-1138.
26. POPOVIC, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14 2120-2148. MR2100386
27. RANNALA, B. (1997). On the genealogy of a rare allele. Theoret. Popul. Biol. 52 216-223.
28. STADLER, T. (2009). On incomplete sampling under birth-death models and connections to the sampling-based coalescent. J. Theoret. Biol. 261 58-66.