On the Recognition and Structure of Probability Generating Functions

  • Anthony G. Pakes
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)


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

Key words.

Generating functions Harmonic renewal functions Infinitely divisible laws Branching processes 


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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Anthony G. Pakes
    • 1
  1. 1.Department of MathematicsUniversity of Western AustraliaNedlandsAustralia

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