Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-18T09:51:09.557Z Has data issue: false hasContentIssue false
  • English
  • Français

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

Published online by Cambridge University Press:  14 July 2016

D. I. Saunders
D. I. Saunders*
Affiliation:
South Australian Institute of Technology, Ingle Farm, South Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

Keywords

AGE-DEPENDENTBRANCHING PROCESSARBITRARY STATE SPACEMALTHUSIAN BEHAVIOUR
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 455 - 465
Copyright
Copyright © Applied Probability Trust 1976 

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] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[2] Bircher, J. J. and Mode, C. J. (1971) An age-dependent branching process with arbitrary state space I. J. Math. Anal. Appl. 36, 41–59.CrossRefGoogle Scholar
[3] Dunford, N. and Schwartz, J. (1959) Linear Operators. Part I, General Theory. Interscience, New York.Google Scholar
[4] Haar, A. (1927) Über asymptotische Entwicklungen von Funktionen. Math. Ann. 96, 69–107.CrossRefGoogle Scholar
[5] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[6] Karlin, S. (1959) Positive operators. J. Math. Mech. 8, 907–937.Google Scholar
[7] Lovitt, W. V. (1950) Linear Integral Equations. Dover, New York.Google Scholar
[8] Mode, C. J. (1971) Multitype Branching Processes. American Elsevier, New York.Google Scholar
[9] Mode, C. J. (1971) Applications of the Fredholm theory in Hilbert space to infinite systems of renewal type integral equations. Math. Biosci. 12, 347–366.CrossRefGoogle Scholar
[10] Mode, C. J. (1972) Limit theorems for infinite systems of renewal type integral equations arising in age-dependent branching processes. Math. Biosci. 13, 165–177.CrossRefGoogle Scholar
[11] Mode, C. J. and Bircher, J. J. (1970) On the foundations of age-dependent branching processes with arbitrary state space. J. Math. Anal. Appl. 32, 435–444.CrossRefGoogle Scholar
[12] Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.CrossRefGoogle Scholar
[13] Nair, K. A. and Mode, C. J. (1971) A multidimensional age-dependent branching process — subcritical case. J. Math. Anal. Appl. 34, 567–577.CrossRefGoogle Scholar
[14] Saunders, D. I. (1975) Branching Processes with Arbitrary Type Space. , Flinders University of South Australia.Google Scholar
[15] Schumitzky, A. and Wenska, T. (1975) An operator residue theorem with applications to branching processes and renewal type integral equations. SIAM J. Math. Anal. 6, 229–235.CrossRefGoogle Scholar
[16] Tricomi, F. G. (1975) Integral Equations. Interscience, New York.Google Scholar
[17] Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton, N.J.Google Scholar