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On the Recognition and Structure of Probability Generating Functions

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Classical and Modern Branching Processes

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 84))

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Abstract

If

$$ M\left( s \right) = 1 - {e^{ - \pi (s)}} $$

is a probability generating function, the coefficients π j in the MacLaurin expansion π(s) comprise a harmonic renewal sequence. A simple sufficient condition is given which ensures that a non-negative sequence is harmonic renewal. This condition covers the case of the limiting conditional law of a subcritical Markov branching process.

Examples are given illustrating the limitation of the criterion. The parallel problem for continuous laws and its relation to the CB-process is discussed.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Western Australia, Nedlands, WA, 6907, Australia

    Anthony G. Pakes

Authors
  1. Anthony G. Pakes
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Iowa State University, Ames, IA, 50011, USA

    Krishna B. Athreya

  2. School of Mathematics and Computing Science, Chalmers University of Technology, Gothenburg University, Gothenburg, S-412 96, Sweden

    Peter Jagers

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© 1997 Springer Science+Business Media New York

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Pakes, A.G. (1997). On the Recognition and Structure of Probability Generating Functions. In: Athreya, K.B., Jagers, P. (eds) Classical and Modern Branching Processes. The IMA Volumes in Mathematics and its Applications, vol 84. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1862-3_21

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  • DOI: https://doi.org/10.1007/978-1-4612-1862-3_21

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7315-8

  • Online ISBN: 978-1-4612-1862-3

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