Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T06:45:10.505Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Fisher, R. A. (1922) On the dominance ratio. Proc. Roy. Soc. Edinburgh, 42, 321–341.CrossRefGoogle Scholar
Fisher, R. A. (1930) The distribution of gene ratios for rare mutations. Proc. Roy. Soc. Edinburgh 50, 205–220.Google Scholar
Haldane, J. B. S. (1939) The equilibrium between mutation and random extinction. Ann. Eugen. 9, 400–405.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer Verlag. Berlin, Göttingen, Heidelberg.CrossRefGoogle Scholar
Jordan, C. (1965) Calculus of Finite Differences. Chelsea Publishing Company, New York.Google Scholar
Kojima, K. and Kelleher, T. M. (1962) Survival of mutant genes. Amer. Naturalist. 96, 329–346.CrossRefGoogle Scholar
Kolmogorov, A. N. (1938) Zur Lösung einer biologischen Aufgabe. Comm. Math. Mech. Chebychev. Univ., Tomsk. 2, 1–6.Google Scholar
Nagaev, S. V. and Muhamedhanova, R. (1966) Transient phenomena in branching stochastic processes with discrete time. (In Russian). Limit Theorems and Statistical Inference, 83–89. Izdat. FAN, Uzhbekskoi S.S.R. Tashkent.Google Scholar
Nagaev, S. V. and Muhamedhanova, R. (1968) Certain remarks apropos of earlier published limit theorems in the theory of branching processes. (In Russian). Probabilistic Models and Quality Control, 46–49. Izdat. FAN, Uzhbekskoi S.S.R. Tashkent.Google Scholar
Seneta, E. (1967) On the transient behavior of a Poisson branching process. J. Aust. Math. Soc. 7, 465–480.CrossRefGoogle Scholar
Seneta, E. (1968) On asymptotic properties of sub-critical branching processes. J. Aust. Math. Soc. 8, 671–682.CrossRefGoogle Scholar
Yaglom, A. M. (1947) Certain limit theorems in the theory of branching stochastic processes. (In Russian). Dokl. Akad. Nauk. S.S.S.R. 56, 795–798.Google Scholar