Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-19T21:23:56.649Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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 

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

[1] Bahadur, R. and Rao, R. R. (1960) On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027.CrossRefGoogle Scholar
[2a] Bühler, W. J. (1970) Generations and degree of relationship in supercritical Markov branching processes. (To be published).Google Scholar
[2b] Bühler, W. J. (1970) The distribution of generations and other aspects of the family structure of branching processes. (To be published).Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II, John Wiley and Sons, New York.Google Scholar
[4] Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Translated from the Russian by Chung, K. L. Addison Wesley, Reading, Mass. Google Scholar
[5] Harris, T. E. (1963) The Theory of Branching Processes. Prentice-Hall, Englewood Cliffs, New Jersey.CrossRefGoogle Scholar
[6] Kharlamov, B. P. (1969) On numbers of particle generations for branching processes with overlapping generations. (In Russian). Teor. Veroyat. Primen. 14, 44–50.Google Scholar
[7] Martin-Löf, A. (1966) A limit theorem for the size of the nth generation of an age-dependent branching process. J. Math. Anal. Appl. 15, 273–279.CrossRefGoogle Scholar
[8] Ney, P. E. (1965) The convergence of a random distribution function associated with a branching process. J. Math. Anal. Appl. 12, 316–327.CrossRefGoogle Scholar
[9] Petrov, V. V. (1965) On the probabilities of large deviations for sums of independent random variables. Theor. Probability Appl. 10, 287–298.CrossRefGoogle Scholar
[10] Takács, L. (1956) On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 32, 488–498.CrossRefGoogle Scholar