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
  • 68 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 

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