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Multitype linear fractional branching processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

A. Joffe  and
G. Letac
A. Joffe*
Affiliation:
Université de Montréal
G. Letac*
Affiliation:
Université Paul Sabatier
*
∗Postal address: Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, QC H3C 3J7, Canada.
∗∗Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex, France. Email address: letac@cict.fr
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Abstract

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We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.

Keywords

Linear fractional birth lawmultitype branching

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1091 - 1106
Copyright
© Applied Probability Trust 2006 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Goryainov, V. V. (1993). Fractional iteration of probability-generating functions and imbedding discrete branching processes into continuous processes. Mat. Sb. 184, 55–74 (in Russian). English translation: Russian Acad. Sci. Sb. Math. 79, 47-61, 1994.Google Scholar
Goryainov, V. V. (1995). Semigroups of probability generating function, and infinitely splittable functions. Theory Random Process. V.1 17, 2–9.Google Scholar
Goryainov, V. V. (2002). Koenigs function and fractional iterates of probability generating functions. Sb. Math. 193, 1009–1025.Google Scholar
Grey, D. R. (1975). Two necessary conditions for embeddability of a Galton–Watson branching process. Math. Proc. Camb. Phil. Soc. 78, 339–343.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Karlin, S. and McGregor, J. (1968a). Embeddability of discrete time simple branching processes into continuous time branching processes. Trans. Amer. Math. Soc. 132, 115–136.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1968b). Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Amer. Math. Soc. 132, 137–145.Google Scholar
Pollak, E. (1974). Survival probabilities and extinction times for some multitype branching processes. Adv. Appl. Prob. 6, 446–462.Google Scholar
Toulouse, P. S. (1999). Thèmes de Probabilités et Statistique. Dunod, Paris.Google Scholar