On the convergence of supercritical general (C-M-J) branching processes
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- Nerman, O. Z. Wahrscheinlichkeitstheorie verw Gebiete (1981) 57: 365. doi:10.1007/BF00534830
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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).