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Total progeny in a multitype critical age dependent branching process with immigration

Published online by Cambridge University Press:  14 July 2016

Howard J. Weiner
Howard J. Weiner*
Affiliation:
University of California, Davis
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Abstract

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

Keywords

MULTITYPE CRITICAL AGE DEPENDENT BRANCHING PROCESSASYMPTOTIC MOMENTSTOTAL PROGENYLIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 3 , September 1974 , pp. 458 - 470
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Jagers, P. (1968) Age dependent branching processes allowing immigration. Theor. Probability Appl. 13, 225–236.Google Scholar
[2] Kaplan, N. (1973) Limit theorems for an age dependent branching process with immigration. To appear.Google Scholar
[3] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[4] Pakes, A. G. (1971) Some limit theorems for the total progeny of a branching process. Adv. Appl. Prob. 3, 176–192.CrossRefGoogle Scholar
[5] Pakes, A. G. (1972) A limit theorem for the integral of a critical age-dependent branching process. Math. Biosci. 13, 109–112.Google Scholar
[6] Weiner, H. (1965) Asymptotic properties of an age-dependent branching process. Ann. Math. Statist. 36, 1565–1568.Google Scholar
[7] Weiner, H. (1966) On age-dependent branching processes. J. Appl. Prob. 3, 383–402.Google Scholar
[8] Weiner, H. (1970) On a multi-type critical age-dependent branching process. J. Appl. Prob. 7, 523–543.Google Scholar
[9] Weiner, H. (1972) A multi-type critical age-dependent branching process with immigration. J. Appl. Prob. 9, 697–706.CrossRefGoogle Scholar
[10] Weiner, H. (1974) Total immigration in a critical age-dependent branching process. Submitted.Google Scholar