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Martingale convergence in the branching random walk

Published online by Cambridge University Press:  14 July 2016

J. D. Biggins
J. D. Biggins*
Affiliation:
University of Oxford
*
*Now at the University of Sheffield.
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Abstract

A result like the Kesten-Stigum theorem is obtained for certain martingales associated with the branching random walk. A special case, when a ‘Malthusian parameter’ exists, is considered in greater detail and a link with some known results about the Crump-Mode model for a population is established.

Keywords

BRANCHING RANDOM WALKSMARTINGALESFUNCTIONAL EQUATIONBRANCHING PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 25 - 37
Copyright
Copyright © Applied Probability Trust 1977 

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References

Athreya, K. B. and Kaplan, N. (1976) Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process. Ann. Prob. 4, 38–50.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Biggins, J. D. (1976) The first-and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.CrossRefGoogle Scholar
Bingham, N. H. and Doney, R. A. (1975) Asymptotic properties of supercritical branching processes 2: Crump-Mode and Jirina processes. Adv. Appl. Prob. 7, 66–82.CrossRefGoogle Scholar
Crump, K. S. and Mode, C. J. (1968) A general age-dependent branching process 1. J. Math. Anal. Appl. 24, 494–508.CrossRefGoogle Scholar
Doney, R. A. (1972) A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.CrossRefGoogle Scholar
Doney, R. A. (1973) On a functional equation for general branching processes. J. Appl. Prob. 10, 198–205.CrossRefGoogle Scholar
Doney, R. A. (1976) On single and multi-type general age-dependent branching processes. J. Appl. Prob. 13, 239–246.CrossRefGoogle Scholar
Erdös, P. (1949) On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Hammersley, J. M. (1974) Postulates for subadditive processes. Ann. Prob. 2, 652–680.CrossRefGoogle Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Kingman, J. F. C. (1975) The first birth problem for an age-dependent branching process. Ann. Prob. 3, 790–801.CrossRefGoogle Scholar
Ney, P. E. (1964) Generalized branching processes 1: existence and uniqueness theorems. Illinois J. Math. 8, 316–331.Google Scholar
Spitzer, F. (1956) A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, 323–339.CrossRefGoogle Scholar