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The first- and last-birth problems for a multitype age-dependent branching process

Published online by Cambridge University Press:  01 July 2016

J. D. Biggins
J. D. Biggins*
Affiliation:
University of Oxford
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Abstract

If Bn is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generation Bn/n converges to a constant; if Dn is the time of the last birth in the nth generation then Dn/n also converges to a constant on the survival set. Analogous results hold for the extreme members of the nth generation in a branching random walk.

Keywords

CRUMP-MODE PROCESSBRANCHING RANDOM WALKFIRST BIRTHLAST BIRTHEXTREMESMULTITYPE
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 3 , September 1976 , pp. 446 - 459
Copyright
Copyright © Applied Probability Trust 1976 

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References

Hammersley, J. M. (1974) Postulates for subadditive processes. Ann. Prob. 2, 652–680.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
Kingman, J. F. C. (1975) The first birth problem for an age-dependent branching process. Ann. Prob. 3, 790–801.Google Scholar
Miller, H. D. (1961) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1260–1270.Google Scholar
Mode, C. J. (1971) Multitype Branching Processes, Elsevier, New York.Google Scholar
Ney, P. E. (1964) Generalized branching processes 1: Existence and uniqueness theorems. Illinois J. Math. 8, 316–331.Google Scholar
Pyke, R. and Schaufele, R. (1966) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.Google Scholar
Seneta, E. (1973) Non-Negative Matrices. Allen and Unwin, London.Google Scholar