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Supercritical Markov branching processes with general set of types

  • H. Hering
Article

Abstract

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].

Keywords

Distribution Function Stochastic Process Asymptotic Behaviour Probability Theory Final Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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