Summary
Convergence in probability of Malthus normed supercritical general branching processes (i.e. Crump-Mode-Jagers branching processes) counted with a general characteristic are established, provided the latter satisfies mild regularity conditions. If the Laplace transform of the reproduction point process evaluated in the Malthusian parameter has a finite ‘x log x-moment’ convergence in probability of the empirical age distribution and more generally of the ratio of two differently counted versions of the process also follow.
Malthus normed processes are also shown to converge a.s., provided the tail of the reproduction point process and the characteristic both satisfy mild regularity conditions. If in addition the ‘x log x-moment’ above is finite a.s. convergence of ratios follow.
Further, a finite expectation of the Laplace-transform of the reproduction point process evaluated in any point smaller than the Malthusian parameter is shown to imply a.s. convergence of ratios even if the ‘x log x-moment’ above equals infinity.
Straight-forward generalizations to the multi-type case are available in Nerman (1979).
Article PDF
Similar content being viewed by others
References
Asmussen, S., Kurtz, T.G.: Necessary and sufficient conditions for complete convergence in the law of large numbers. Ann. Probability 8, 176–182 (1980)
Athreya, K.B.: On the supercritical agedependent branching process. Ann. Math. Statist. 40, 743–763 (1969)
Athreya, K.B., Kaplan, N.: Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process. Ann. Probability 4, 38–50 (1976)
Athreya, K.B., Kaplan, N.: The additive property and its applications in branching processes. Adv. in Probability vol 5, branching processes (1978)
Bauer, H.: Wahrscheinlichkeitstheorie und Grundzüge der Massteorie. Berlin: de Gruyter, 1968
Breiman, L.: Probability. Reading, Massachusetts: Addison Wesley, 1968
Doney, R.A.: A limit theorem for a class of supercritical branching processes. J. Appl. Probability 9, 707–724 (1972)
Doney, R.A.: On single- and multi-type general age-dependent branching processes. J. Appl. Probability 13, 239–246 (1976)
Härnqvist, M.: Limit theorems for point processes generated in a general branching process. To appear in Adv. Appl. Prob. (1981)
Jagers, P.: Branching Processes with Biological Applications. London: Wiley, 1975
Jagers, P.: How probable is it to be first-born? And other branching process applications to kinship problems. To appear in Mathematical Biosciences (1981)
Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 1211–1223 (1966)
Kuczek, T.: On the convergence of the empiric age distribution for one dimensional super-critical age dependent branching processes. Rutgers University, New Brunswick, New Jersey (1980). To appear in Ann. Probability
Meyer, P.-A.: Martingales and Stochastic Integrals I. Berlin Heidelberg New York: Springer, 1972
Nerman, O.: On the Convergence of Supercritical General Branching Processes. Thesis, Department of Mathematics, Chalmers University of Technology and the University of Göteborg 1979
Neveu, J.: Discrete-Parameter Martingales. Amsterdam: North-Holland, 1975
Rama-Murthy, K.: Convergence of State Distributions in Multitype Bellman-Harris and Crump-Mode-Jagers Branching Processes. Thesis, Department of Appl. Math. Bangalore: Indian Institute of Science, 1978
Savits, T.H.: The Supercritical Multi-Type Crump and Mode Age-dependent Model. Unpublished manuscript, University of Pittsburgh, 1975
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nerman, O. On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 365–395 (1981). https://doi.org/10.1007/BF00534830
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00534830