Multitype branching processes and Feller diffusion processes are conditioned on very late extinction. The conditioned laws are expressed as Doob h-transforms of the unconditioned laws, and an interpretation of the conditioned paths for the branching process is given, via the immortal particle. We study different limits for the conditioned process (increasing delay of extinction, long-time behavior, scaling limit) and provide an exhaustive list of exchangeability results.
Mots clés : multitype branching process, Feller diffusion process, conditioned branching process, diffusion limit, extinction, immortal particle, long-time behavior
@article{PS_2011__15__417_0,
author = {P\'enisson, Sophie},
title = {Continuous-time multitype branching processes conditioned on very late extinction},
journal = {ESAIM: Probability and Statistics},
pages = {417--442},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2010011},
mrnumber = {2870524},
zbl = {1278.60134},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2010011/}
}
TY - JOUR AU - Pénisson, Sophie TI - Continuous-time multitype branching processes conditioned on very late extinction JO - ESAIM: Probability and Statistics PY - 2011 SP - 417 EP - 442 VL - 15 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2010011/ DO - 10.1051/ps/2010011 LA - en ID - PS_2011__15__417_0 ER -
%0 Journal Article %A Pénisson, Sophie %T Continuous-time multitype branching processes conditioned on very late extinction %J ESAIM: Probability and Statistics %D 2011 %P 417-442 %V 15 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2010011/ %R 10.1051/ps/2010011 %G en %F PS_2011__15__417_0
Pénisson, Sophie. Continuous-time multitype branching processes conditioned on very late extinction. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 417-442. doi : 10.1051/ps/2010011. http://archive.numdam.org/articles/10.1051/ps/2010011/
[1] and , Branching Processes. Springer-Verlag (1972). | MR | Zbl
[2] and , Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions. Electronic Journal of Probability 13 (2008) 777-810. | MR | Zbl
[3] and , The -process in a multitype branching process. Int. J. Pure Appl. Math. 42 (2008) 235-240. | MR | Zbl
[4] and , Markov processes: characterization and convergence. Wiley (1986). | MR | Zbl
[5] , Two representations of a conditioned superprocess, in Proc. R. Soc. Edinb. Sect. A 123 (1993) 959-971. | MR | Zbl
[6] , Diffusion processes in genetics, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles (1951) 227-246. | MR | Zbl
[7] , Matrizentheorie. Springer-Verlag (1986). | MR
[8] and , Supercritical multitype branching processes: the ancestral types of typical individuals. Adv. Appl. Probab. 35 (2003) 1090-1110. | MR | Zbl
[9] and , Ein Grenzwertsatz für subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen. Math. Nachr. 64 (1974) 357-362. | MR | Zbl
[10] and , The structure of reduced critical Galton-Watson processes. Math. Nachr. 79 (1977) 233-241. | MR | Zbl
[11] and , General branching processes conditioned on extinction are still branching processes. Electronic Communications in Probability 13 (2008) 540-547. | MR | Zbl
[12] and , Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Probab. 18 (1986) 20-65. | MR | Zbl
[13] and , On multitype branching processes with . J. Math. Anal. Appl. 19 (1967) 409-430. | MR | Zbl
[14] and , Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 (1971) 34-51. | MR | Zbl
[15] , Zur Lösung einer biologischen Aufgabe. Comm. Math. Mech. Chebyshev Univ. Tomsk 2 (1938). | Zbl
[16] , Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electronic Journal of Probability 12 (2007) 420-446. | MR | Zbl
[17] and , Conditioned branching process and their limiting diffusions. Theory Probab. Appl. 13 (1968) 126-137. | MR | Zbl
[18] , Asymptotic behavior of multitype Galton-Watson processes. J. Math. Kyoto Univ. 15 (1975) 251-302. | MR | Zbl
[19] and , Processus de Dawson-Watanabe conditionné par le futur lointain. C. R. Acad. Sci. Sér. I Math. 309 (1989) 867-872. | MR | Zbl
[20] , Non-negative matrices - An introduction to theory and applications. Halsted Press (1973). | MR | Zbl
[21] , Verzweigungsprozesse. R. Oldenbourg Verlag (1975). | MR | Zbl
[22] , Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795-798. | MR | Zbl
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