Acta Applicandae Mathematicae

, Volume 138, Issue 1, pp 279–312 | Cite as

A Two-Type Bellman–Harris Process Initiated by a Large Number of Particles

  • Vladimir Vatutin
  • Alexander Iksanov
  • Valentin Topchii
Article
  • 76 Downloads

Abstract

We investigate a two-type critical Bellman–Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order o(t −2); the tail of the life-length distribution of the second type particles is regularly varying at infinity with index −β, β∈(0,1]; at time t=0 the process starts with a large number N of the second type particles and no particles of the first type. It is shown that the time axis 0≤t<∞ splits into several regions whose ranges depend on β and the ratio N/t within each of which the process at time t exhibits asymptotics (as N,t→∞) which is different from those in the other regions.

Keywords

Evolutionary stages Two-type critical Bellman–Harris process Regular variation Renewal theory 

Notes

Acknowledgement

The research was supported in part by the Programs of RAS “Dynamical Systems and Control Theory” (V. Vatutin) and “Development of the methods for investigating some stochastic models aimed towards population and biomedical applications” (V. Topchii). The authors thank two anonymous referees for making a number of valuable remarks that helped improving the presentation of this work.

References

  1. 1.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. In: Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1989). Paperback ed. 512 p. Google Scholar
  2. 2.
    Chistyakov, V.P.: The asymptotic behavior of the non-extinction probability for a critical branching process. Theory Probab. Appl. 16, 620–630 (1971) MATHCrossRefGoogle Scholar
  3. 3.
    Sevast’yanov, B.A.: Verzweigungsprozesse. Akademie Verlag, Berlin (1974). xi+326 pp. MATHGoogle Scholar
  4. 4.
    Shurenkov, V.M.: Two limit theorems for critical branching processes. Theory Probab. Appl. 21, 534–544 (1977) CrossRefGoogle Scholar
  5. 5.
    Vatutin, V.A.: Discrete limit distributions of the number of particles in a Bellman–Harris branching process with several types of particles. Theory Probab. Appl. 24, 509–520 (1979) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Vatutin, V.A.: Critical Bellman–Harris branching processes starting with a large number of particles. Math. Notes 40, 803–811 (1986) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Vatutin, V.A., Topchii, V.A.: Critical Bellman–Harris branching processes with long living particles. Proc. Steklov Inst. Math. 282, 243–272 (2013) MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Vladimir Vatutin
    • 1
  • Alexander Iksanov
    • 2
  • Valentin Topchii
    • 3
  1. 1.Steklov Mathematical institute RASMoscowRussia
  2. 2.Faculty of CyberneticsNational Taras Shevchenko University of KyivKyivUkraine
  3. 3.Sobolev Institute of Mathematics SB RASNovosibirskRussia

Personalised recommendations