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Distribution of the branching-process population among generations

Published online by Cambridge University Press:  14 July 2016

M. L. Samuels
M. L. Samuels*
Affiliation:
Purdue University, Lafayette, Indiana
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Summary

In a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).

Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.

For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 655 - 667
Copyright
Copyright © Applied Probability Trust 1971 

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References

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