Decomposition of Supercritical Linear-Fractional Branching Processes

Affiliation(s)

Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.

Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan.

Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.

Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan.

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.

Cite this paper

S. Sagitov and A. Shaimerdenova, "Decomposition of Supercritical Linear-Fractional Branching Processes,"*Applied Mathematics*, Vol. 4 No. 2, 2013, pp. 352-359. doi: 10.4236/am.2013.42054.

S. Sagitov and A. Shaimerdenova, "Decomposition of Supercritical Linear-Fractional Branching Processes,"

References

[1] C. C. Heyde and E. J. Seneta “Bienayme: Statistical Theory Anticipated,” Springer, New York, 1977. doi:10.1007/978-1-4684-9469-3

[2] B. A. Sevastyanov, “The Theory of Branching Random Processes,” Uspehi Matematicheskih Nauk, Vol. 6, 1951, pp. 47-99.

[3] B. A. Sewastjanow, “Verzweigungsprozesse,” AkademieVerlag, Berlin, 1974.

[4] 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

[5] K. B. Athreya and P. E. Ney, “Branching Processes,” Dover, Mineola, 2004.

[6] 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

[7] S. Sagitov, “Linear-Fractional Branching Processes with Countably Many Types,” 2012, 24 p. http://arxiv.org/abs/1111.4689

[8] 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

[1] C. C. Heyde and E. J. Seneta “Bienayme: Statistical Theory Anticipated,” Springer, New York, 1977. doi:10.1007/978-1-4684-9469-3

[2] B. A. Sevastyanov, “The Theory of Branching Random Processes,” Uspehi Matematicheskih Nauk, Vol. 6, 1951, pp. 47-99.

[3] B. A. Sewastjanow, “Verzweigungsprozesse,” AkademieVerlag, Berlin, 1974.

[4] 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

[5] K. B. Athreya and P. E. Ney, “Branching Processes,” Dover, Mineola, 2004.

[6] 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

[7] S. Sagitov, “Linear-Fractional Branching Processes with Countably Many Types,” 2012, 24 p. http://arxiv.org/abs/1111.4689

[8] 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