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Decomposition of Supercritical Linear-Fractional Branching Processes
Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden
Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan
- 1 Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden
- 2 Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan
Applied Mathematics·Volume 04 (2013)·Pages 352–359·Published 22 February 2013·DOI10.4236/am.2013.42054
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Abstract
It is well known that a supercritical single-type Bienayme-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienayme-Galton-Watson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.
KeywordsHarris-Sevastyanov TransformationDual Reproduction LawBranching Process with Countably Many TypesMultivariate Linear-Fractional DistributionBienaymé-Galton-Watson ProcessConditioned Branching Process
- C. C. Heyde and E. J. Seneta “Bienayme: Statistical Theory Anticipated,” Springer, New York, 1977. doi:10.1007/978-1-4684-9469-3
- B. A. Sevastyanov, “The Theory of Branching Random Processes,” Uspehi Matematicheskih Nauk, Vol. 6, 1951, pp. 47-99.
- B. A. Sewastjanow, “Verzweigungsprozesse,” AkademieVerlag, Berlin, 1974.
- N. O’Connell, “Yule Process Approximation of the Skeleton of a Branching Process,” Journal of Applied Probability, Vol. 30, No. 3, 1993, pp. 725-729. doi:10.2307/3214778
- K. B. Athreya and P. E. Ney, “Branching Processes,” Dover, Mineola, 2004.
- F. Klebaner, U. Rosler and S. Sagitov, “Transformations of Galton-Watson Processes and Linear Fractional Reproduction,” Advances in Applied Probability, Vol. 39, No. 4, 2007, pp. 1036-1053. doi:10.1239/aap/1198177238
- S. Sagitov, “Linear-Fractional Branching Processes with Countably Many Types,” 2012, 24 p. http://arxiv.org/abs/1111.4689
- P. Jagers and A. N. Lager?s, “General Branching Processes Conditioned on Extinction Are Still Branching Processes,” Electronic Communications in Probability, Vol. 13, 2008, pp. 540-547. doi:10.1214/ECP.v13-1419