Continuous-time multitype branching processes conditioned on very late extinction***

Sophie Pénisson

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 417-442
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Pénisson, Sophie. "Continuous-time multitype branching processes conditioned on very late extinction***." ESAIM: Probability and Statistics 15 (2012): 417-442. <http://eudml.org/doc/222470>.

@article{Pénisson2012,
abstract = { 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. },
author = {Pénisson, Sophie},
journal = {ESAIM: Probability and Statistics},
keywords = {Multitype branching process; Feller diffusion process; conditioned branching process; diffusion limit; extinction; immortal particle; long-time behavior; multitype branching process; Feller diffusion},
language = {eng},
month = {1},
pages = {417-442},
publisher = {EDP Sciences},
title = {Continuous-time multitype branching processes conditioned on very late extinction***},
url = {http://eudml.org/doc/222470},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Pénisson, Sophie
TI - Continuous-time multitype branching processes conditioned on very late extinction***
JO - ESAIM: Probability and Statistics
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 417
EP - 442
AB - 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.
LA - eng
KW - Multitype branching process; Feller diffusion process; conditioned branching process; diffusion limit; extinction; immortal particle; long-time behavior; multitype branching process; Feller diffusion
UR - http://eudml.org/doc/222470
ER -

References

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  1. K.B. Athreya and P.E. Ney, Branching Processes. Springer-Verlag (1972).  
  2. N. Champagnat and S. Rœlly, Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions. Electronic Journal of Probability13 (2008) 777–810.  Zbl1189.60154
  3. S. Dallaporta and A. Joffe, The Q -process in a multitype branching process. Int. J. Pure Appl. Math.42 (2008) 235–240.  Zbl1136.60360
  4. S.N. Ethier and T.G. Kurtz, Markov processes: characterization and convergence. Wiley (1986).  Zbl0592.60049
  5. S.N. Evans, Two representations of a conditioned superprocess, in Proc. R. Soc. Edinb. Sect. A123 (1993) 959–971.  Zbl0784.60052
  6. W. Feller, 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.  
  7. F.R. Gantmacher, Matrizentheorie. Springer-Verlag (1986).  
  8. H.O. Georgii and E. Baake, Supercritical multitype branching processes: the ancestral types of typical individuals. Adv. Appl. Probab.35 (2003) 1090–1110.  Zbl1044.60080
  9. K. Fleischmann and U. Prehn, Ein Grenzwertsatz für subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen. Math. Nachr.64 (1974) 357–362.  Zbl0278.60055
  10. K. Fleischmann and R. Siegmund-Schultze, The structure of reduced critical Galton-Watson processes. Math. Nachr.79 (1977) 233–241.  Zbl0299.60065
  11. P. Jagers and A.N. Lagerås, General branching processes conditioned on extinction are still branching processes. Electronic Communications in Probability13 (2008) 540–547.  Zbl1189.60158
  12. A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Probab.18 (1986) 20–65.  Zbl0595.60008
  13. A. Joffe and F. Spitzer, On multitype branching processes with ρ 1 . J. Math. Anal. Appl.19 (1967) 409–430.  Zbl0178.19504
  14. K. Kawazu and S. Watanabe, Branching processes with immigration and related limit theorems. Theory Probab. Appl.16 (1971) 34–51.  Zbl0242.60034
  15. A.N. Kolmogorov, Zur Lösung einer biologischen Aufgabe. Comm. Math. Mech. Chebyshev Univ. Tomsk2 (1938).  Zbl0019.35901
  16. A. Lambert, Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electronic Journal of Probability12 (2007) 420–446.  Zbl1127.60082
  17. J. Lamperti and P. Ney, Conditioned branching process and their limiting diffusions. Theory Probab. Appl.13 (1968) 126–137.  Zbl0253.60073
  18. Y. Ogura, Asymptotic behavior of multitype Galton-Watson processes. J. Math. Kyoto Univ.15 (1975) 251–302.  Zbl0314.60060
  19. S. Rœlly and A. Rouault, Processus de Dawson-Watanabe conditionné par le futur lointain. C. R. Acad. Sci. Sér. I Math.309 (1989) 867–872.  Zbl0684.60062
  20. E. Seneta, Non-negative matrices – An introduction to theory and applications. Halsted Press (1973).  Zbl0278.15011
  21. B.A. Sewastjanow, Verzweigungsprozesse. R. Oldenbourg Verlag (1975).  Zbl0344.60049
  22. A.M. Yaglom, Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.)56 (1947) 795–798.  

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