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On survival probabilities and extinction times for some branching processes

Published online by Cambridge University Press:  14 July 2016

Edward Pollak
Edward Pollak*
Affiliation:
Iowa State University, Ames, Iowa
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Summary

We consider branching processes for which the first three moments of the distribution of offspring exist. Let f(t) andz be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f(t) = t. Then if M = f'(z) and fn(t) is equal to the generating function of the distribution of nth generation descendants of a single individual, it is known that, quite generally, (z – fn(0)) / Mn tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even ifzis less than 1. Numerical values are given for the Poisson and negative binomial cases.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 633 - 654
Copyright
Copyright © Applied Probability Trust 1971 

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