Probability Theory and Related Fields

, Volume 87, Issue 2, pp 209–240 | Cite as

Markov branching processes with instantaneous immigration

  • A. Y. Chen
  • E. Renshaw
Article

Summary

Markov branching processes with instantaneous immigration possess the property that immigration occurs immediately the number of particles reaches zero, i.e. the conditional expectation of sojourn time at zero is zero. In this paper we consider the existence and uniqueness of such a structure. We prove that if the sum of the immigration rates is finite then no such structure can exist, and we provide a necessary and sufficient condition for existence for the case in which this sum is infinite. Study of the uniqueness problem shows that for honest processes the solution is unique.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Athreya, K.B., Ney, P.E.: Branching processes. Berlin Heidelberg New York: Springer 1972Google Scholar
  2. 2.
    Chung, K.L.: Markov chains with stationary transition probabilities. Berlin Heidelberg New York: Springer 1967Google Scholar
  3. 3.
    Chung, K.L.: Lectures on boundary theory for Markov chains. Ann. Math. Stud.65 (1970)Google Scholar
  4. 4.
    Doney, R.A.: A note on some results of Schuh. J. Appl. Probab.21, 192–196 (1984)Google Scholar
  5. 5.
    Doob, J.L.: Markoff chains-denumerable case. Trans. Am. Math. Soc.58, 455–473 (1945)Google Scholar
  6. 6.
    Doob, J.L.: Stochastic processes. New York: Wiley 1953Google Scholar
  7. 7.
    Feller, W.: On the integro-differential equations of purely discontinuous Markov processes. Trans. Am. Math. Soc.48, 488–515 (1940)Google Scholar
  8. 8.
    Foster, J.H.: A limit theorem for a branching process with state-dependent immigration. Ann. Math. Statist.42, 1773–1776 (1971)Google Scholar
  9. 9.
    Freedman, D.: Markov chains. Berlin Heidelberg New York: Springer 1983Google Scholar
  10. 10.
    Harris, T.E.: The theory of branching processes. Berlin Heidelberg New York: Springer 1963Google Scholar
  11. 11.
    Hille, E.: Functional analysis and semi-groups. Colloq. Publ., Am. Math. Soc. (1948)Google Scholar
  12. 12.
    Kemeny, J.G., Snell, J., Knapp, A.W.: Denumerable Markov chains. Princeton: Van Nostrand 1966Google Scholar
  13. 13.
    Kendall, D.G.: Some analytical properties of continuous stationary Markov transition functions. Trans. Am. Math. Soc.78, 529–540 (1955)Google Scholar
  14. 14.
    Kendall, D.G., Reuter, G.E.H.: Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators onl. Proc. Intern. Congr. Math. Amsterdam, Vol. III, pp. 377–415 (1954)Google Scholar
  15. 15.
    Kolmogorov, A.N.: On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states. Moskov. Gos. Univ. Učenye Zapiski Matematika.148, 53–59 (1951)Google Scholar
  16. 16.
    Lévy, P.: Complément à l'étude des processus de Markoff. Ann. Sci. Éc. Norm. Supér. (3),69, 203–212 (1952)Google Scholar
  17. 17.
    Mitov, K.V., Vatutin, V.A., Yanev, N.M.: Continuous-time branching processes with decreasing state-dependent immigration. Adv. Appl. Probab.16, 697–714 (1984)Google Scholar
  18. 18.
    Neveu, J.: Lattice methods and submarkovian processes. Proc. 4th Berk. Symp. Math. Statist. Prob., Vol 2, pp. 347–391. University of California Press (1960)Google Scholar
  19. 19.
    Pakes, A.G.: A branching process with a state-dependent immigration component. Adv. Appl. Probab.3, 301–314 (1971)Google Scholar
  20. 20.
    Pakes, A.G.: Some results for non-supercritical Galton-Watson processes with immigration. Math. Biosci. 24, 71–92 (1975)Google Scholar
  21. 21.
    Pakes, A.G.: On the age distribution of a Markov chain. J. Appl. Probab.15, 65–77 (1978)Google Scholar
  22. 22.
    Reuter, G.E.H.: Denumerable Markov processes and the associated semigroup onl. Acta. Math.97, 1–46 (1957)Google Scholar
  23. 23.
    Reuter, G.E.H.: Denumerable Markov processes (II). J. Lond. Math. Soc.34, 81–91 (1959)Google Scholar
  24. 24.
    Reuter, G.E.H.: Denumerable Markov processes (III). J. Lond. Math. Soc.37, 63–73 (1962)Google Scholar
  25. 25.
    Reuter, G.E.H.: Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitstheor. Verw. Geb.13, 315–320 (1969)Google Scholar
  26. 26.
    Schuh, H.J.: Sums of i.i.d. random variables and an application to the explosion criterion for Markov branching processes. J. Appl. Probab.19, 29–38 (1982)Google Scholar
  27. 27.
    Seneta, E.: Regularly varying functions. (Lect. Notes Math. vol. 508) Berlin Heidelberg New York: Springer 1976Google Scholar
  28. 28.
    Williams, D.: A note on theQ-matrices of Markov chains. Z. Wahrscheinlichkeitstheor. Verw. Geb.7, 116–121 (1967)Google Scholar
  29. 29.
    Williams, D.: Diffusions, Markov processes and martingales, vol. 1. Foundations. New York: Wiley 1979Google Scholar
  30. 30.
    Yamazato, M.: Some results on continuous time branching processes with state-dependent immigration. J. Math. Soc. Japan,27, 479–496 (1975)Google Scholar
  31. 31.
    Yosida, K.: On the differentiability and the representation of one-parameter semi-groups of linear operators. J. Math. Soc. Japan,1, 15–21 (1948)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • A. Y. Chen
    • 1
  • E. Renshaw
    • 1
  1. 1.Department of Statistics, James Clerk Maxwell Building, King's BuildingsUniversity of EdinburghEdinburghUK

Personalised recommendations