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Limit Theorems for Branching Processes with Random Migration Stopped at Zero

  • George P. Yanev
  • Nickolay M. Yanev
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

In this paper, we study a generalization of the classical BienayméGalton-Watson branching process by allowing a random migration component stopped at zero (i.e. the state zero is absorbing). In each generation for which the population size is positive, with probability p two types of emigration are available - a random number of offsprings and a random number of families (these two random variables can be dependent); with probability q there is not any migration; with probability r an immigration of new individuals is possible, p+q+r = 1. The critical case is investigated with an extension when the initial law is attracted to a stable (p) law, p ≤ 1. The asymptotic form of the probability of non-extinction is studied and conditional limit theorems for the population size are obtained, depending on the range of an additional parameter of criticality.

Key words

random migration extinction moments conditional limit theorems 

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References

  1. Abramowitz, M. and Stegun, I.A. (1970) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York.Google Scholar
  2. Athreya, K.B. and Ney, P.E. (1972) Branching Processes. Springer-Verlag, Berlin. Feller W. (1971) An Introduction to Probability Theory and its Applications, Vol 2. Wiley, New York.Google Scholar
  3. Grey, D.R. (1988) Supercritical branching processes with density independent catastrophes. Math. Proc. Camb. Phil. Soc. 104, 413–416.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Ivanoff, B.G. and Seneta, E. (1985) The critical branching process with immigration stopped at zero. J. Appl. Prob. 22, 223–227.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Kaverin, S.V. (1990) A refinement of limit theorems for critical branching processes with an emigration. Theory Prob. Appl., 35, 570–575.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Mitov, K.V. (1990) Critical branching process with migration stopped at zero. Mathematics and Education in Mathematics, Proc. 19th Spring Conf. UBM, Publ. House of BAS, Sofia, 367–371 (In Bulgarian).Google Scholar
  7. Pakes, A.G. (1975) Some Results for Non-Supercritical Galton-Watson Processes with Immigration. Mathematical Biosciences 24, 71–92.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Seneta, E. and Tavaré, S. (1983) A note on models using the branching processes with immigration stopped at zero. J. Appl. Prob. 20, 11–18.zbMATHCrossRefGoogle Scholar
  9. Sevastyanov (1971) Branching Processes, Nauka, Moscow (In Russian).Google Scholar
  10. Vatutin, V.A. (1977a) A critical Galton-Watson branching process with emigration. Theory Prob. Appl. 22, 482–497.MathSciNetGoogle Scholar
  11. Vatutin, V.A. (1977b) A conditional limit theorem for a critical branching process with immigration. Mathematicheskie Zametki 21, 727–736 (In Russian).Google Scholar
  12. Yanev G.P. and Yanev N.M. (1995a) Critical Branching Processes with Random Migration. In Branching processes. Proceedings of the First World Congress. Lecture Notes in Statistics, Vol. 99, Springer-Verlag, New York, 36–46.Google Scholar
  13. Yanev, G.P. and Yanev N.M. (1995b) Critical Branching Processes in which Emigration Dominates Immigration. (submitted).Google Scholar
  14. Yanev, N.M., Vatutin, V.A. and Mitov, K.V. (1986) Critical branching processes with random migration stopped at zero. Mathematics and Education in Mathematics, Proc. 15th Spring Conf. UBM, Publ. House of BAS, Sofia, 511–517 (In Russian).Google Scholar
  15. Zubkov, A.M. (1972) Life-periods of a branching process with immigration. TheoryProb. Appl., 17, 179–188.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • George P. Yanev
    • 1
  • Nickolay M. Yanev
    • 1
  1. 1.Department of Probability and Statistics, Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria

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