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Series expansions of probability generating functions and bounds for the extinction probability of a branching process

Published online by Cambridge University Press:  14 July 2016

D. J. Daley  and
Prakash Narayan
D. J. Daley*
Affiliation:
The Australian National University
Prakash Narayan*
Affiliation:
Monash University
*
∗Postal address: Statistics Department (IAS), The Australian National University, P.O. Box 4, Canberra, ACT 2600, Australia.
∗∗Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.
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Abstract

In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established.

Keywords

PROBABILITY GENERATING FUNCTIONSFACTORIAL MOMENTSBRANCHING PROCESS EXTINCTION PROBABILITY BOUNDS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 939 - 947
Copyright
Copyright © Applied Probability Trust 

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