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Multidimensional age-dependent branching processes allowing immigration: The limiting distribution

Published online by Cambridge University Press:  14 July 2016

Norman Kaplan*
Affiliation:
University of California, Berkeley

Abstract

This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer, Berlin-Gottingen-Heidelberg.CrossRefGoogle Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. 2. Wiley, New York.Google Scholar
[3] Goldstein, M. J. (1971) Critical age-dependent branching processes: Single and multitype. Z. Wahrscheinlichkeitsth. 17, 7488.CrossRefGoogle Scholar
[4] Heathcote, C. R. (1965) Corrections and comments on the paper “A branching process involving immigration”. J. R. Statist. Soc. B 27, 213217.Google Scholar
[5] Jagers, P. (1968) Age-dependent branching processes allowing immigration. Theor. Probability Appl. 13, 225236.CrossRefGoogle Scholar
[6] Kaplan, N. L. (1973) Multitype Galton Watson process with immigration. Ann. of Probability. 1, 947953.CrossRefGoogle Scholar
[7] Kaplan, N. L. and Pakes, A. G. (1974) Supercritical age-dependent branching processes allowing immigration. Submitted to Stochastic Proc. Appl. CrossRefGoogle Scholar
[8] Karlin, S. (1969) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[9] Katz, M. (1963) The probability in the tail of a distribution. Ann. Math. Statist. 34, 312318.CrossRefGoogle Scholar
[10] Mode, C. J. (1968) A multidimensional age-dependent branching process with applica lions to natural selection I. Math. Biosci. 3, 118.CrossRefGoogle Scholar
[11] Quine, M. P. (1970) The multitype Galton-Watson process with immigration. J. Appl. Prob. 7, 411422.CrossRefGoogle Scholar
[12] Ryan, T. A. (1968) On age-dependent branching processes. Ph. D. dissertation, Cornell University.Google Scholar
[13] Pakes, A. G. and Kaplan, N. L. (1974) On the subcritical Bellman-Harris process with immigration. J. Appl. Prob. 11, No. 4. To appear.CrossRefGoogle Scholar