Supercritical Markov branching processes with general set of types

  • H. Hering


This paper concerns the asymptotic behaviour of normalized averaging processes associated with a supercritical, indecomposable Markov branching process. Results wellknown in case of a finite set of types are extended to processes with an arbitrary set of types. The process parameter is allowed to be discrete or continuous.

Convergence in the quadratic mean is proved on the basis of a weak form of positive regularity. In this setting, limit variables corresponding to different averaging functions are proportional almost everywhere. The rate of convergence is such that process skeletons, defined by uniform partitions of the parameter set, converge with probability 1. Starting from the almost sure convergence of skeletons, we obtain almost sure convergence of the processes themselves. The final sections deals with properties of the limiting distribution functions, in particular with the possible jump at the origin and the existence of a continuous density everywhere else.

Other investigations of supercritical Markov branching processes with an infinite or arbitrary set of types are to be found in [2, 3, 4, 6, 7, 13, 15, 19, 21, 23], and [24].


Distribution Function Stochastic Process Asymptotic Behaviour Probability Theory Final Section 
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  1. 1.
    Athreya,K. B.: Some results on multitype continuous time Markov branching processes. Ann. Math. Statistics 39, 347–357 (1968).Google Scholar
  2. 2.
    Conner,H. E.: Limiting behaviour for age- and position-dependent branching processes. J. Math. Analysis Appl. 13, 265–295 (1966).Google Scholar
  3. 3.
    Conner,H. E.: Asymptotic behaviour of averaging processes for a branching process of restricted Brownian particles. J. Math. Analysis Appl. 20, 466–479 (1967).Google Scholar
  4. 4.
    Davis,A. W.: Branching diffusion processes with no absorbing boundary I, II. J. Math. Analysis Appl. 18, 276–296; 19, 1–25 (1967).Google Scholar
  5. 5.
    Doob,J.L.: Stochastic processes. New York-London-Sidney: Wiley 1953.Google Scholar
  6. 6.
    Harris,T.E.: A theorem on general branching processes. Amer. Math. Soc. Notices 6, 55 (1959).Google Scholar
  7. 7.
    Harris,T.E.: The theory of branching processes. Berlin-Göttingen-Heidelberg: Springer 1963.Google Scholar
  8. 8.
    Hering,H.: Critical Markov branching processes with general set of types. Trans. Amer. Math. Soc. 160, 185–202 (1971).Google Scholar
  9. 9.
    Hille,E., Phillips,R.S.: Functional analysis and semi-groups. 2nd ed. Providence R.I.: Colloq. Publ. Amer. Math. Soc. 1957.Google Scholar
  10. 10.
    Ikeda,N., Nagasawa,M., Watanabe,S.: Branching Markov processes I, II, III. J. Math. Kyoto Univ. 8, 233–278, 365–410 (1968); 9, 95–160 (1969).Google Scholar
  11. 11.
    Karlin,S.: Positive operators. J. Math. Mech. 8, 907–938 (1959).Google Scholar
  12. 12.
    Kesten,H., Stigum,B.: A limit theorem for multi-dimensional Galton-Watson processes. Ann. Math. Statistics 37, 1211–1223 (1966).Google Scholar
  13. 13.
    Kharlamov,B.P.: On properties of branching processes with an arbitrary set of particle types. Theor. Probab. Appl. 13, 84–98 (1968).Google Scholar
  14. 14.
    Kre<in,M. G., Rutman,M.A.: Linear operators leaving invariant a cone in a Banach space. Uspeki Matem. Nauk 3, 1–95 (1948) [Russian]: Amer. Math. Soc. Translat. (1) 26 (1950).Google Scholar
  15. 15.
    Moy, Shu-teh C.: Extension of a limit theorem of Everett, Ulam, and Harris on multitype branching processes to a branching process with countably many types. Ann. Math. Statistics 38, 992–999 (1967).Google Scholar
  16. 16.
    Moyal,J.E.: Multiplicative population chains. Proc. Roy. Soc. London, Ser. A 266, 518–526 (1962).Google Scholar
  17. 17.
    Moyal,J.E.: The general theory of stochastic population processes. Acta Math. (Stockholm) 108, 1–31 (1962).Google Scholar
  18. 18.
    Moyal,J.E.: Multiplicative population processes. J. Appl. Probab. 1, 267–283 (1964).Google Scholar
  19. 19.
    Mullikin,T.W.: Neutron branching processes. J. Math. Analysis Appl. 3, 509–525 (1961).Google Scholar
  20. 20.
    Mullikin,T.W.: Limiting distributions for critical multitype branching processes with discrete time. Trans. Amer. Math. Soc. 106, 469–494 (1963).Google Scholar
  21. 21.
    Sevastyanov,B.A.: Branching stochastic processes for particles diffusion in a restricted domain with absorbing boundaries. Theor. Probab. Appl. 3, 111–126 (1958).Google Scholar
  22. 22.
    Yosida,K., Kakutani,S.: Operator treatment of Markov's processes and mean ergodic theorem. Ann. of Math. (2) 42, 188–218 (1941).Google Scholar
  23. 23.
    Watanabe,S.: On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto Univ. 4, 385–398 (1965).Google Scholar
  24. 24.
    Watanabe,S.: Limit theorems for a class of branching processes. Markov Processes and Potential Theory. Symp. ed. J. Chover. New York-London: Wiley 1967.Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • H. Hering
    • 1
  1. 1.Institut für Mathematische StatistikUniversitÄt Karlsruhe (TH)KarlsruheFederal Republic of Germany

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