Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T06:32:55.198Z Has data issue: false hasContentIssue false
  • English
  • Français

The branching diffusion with immigration

Published online by Cambridge University Press:  14 July 2016

B. G. Ivanoff
B. G. Ivanoff*
Affiliation:
Université de Montréal
Get access Rights & Permissions [Opens in a new window]

Abstract

The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem.

Keywords

RANDOM FIELDRANDOM MEASUREPROBABILITY GENERATING FUNCTIONALCUMULANT AND MOMENT DENSITY FUNCTIONSBRILLINGER'S MIXING CONDITIONSIMPLE BRANCHING DIFFUSION PROCESSBRANCHING DIFFUSION WITH IMMIGRATIONSTEADY-STATE RANDOM FIELD
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 1 - 15
Copyright
Copyright © Applied Probability Trust 1980 

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

Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Brillinger, D. R. (1975) Statistical inference for stationary point processes. In Stochastic Processes and Related Topics, ed. Puri, M. L., Academic Press, New York, 55–99.Google Scholar
Crow, J. F. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P. A. W., Wiley, New York, 299–383.Google Scholar
Dawson, D. A. and Ivanoff, B. G. (1978) Branching diffusions and random measures. In Advances in Probability 5, ed. Ney, P., Dekker, New York, 61–104.Google Scholar
Fisher, L. (1972) A survey of the theory of multidimensional point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, Wiley, New York.Google Scholar
Fleischman, J. (1978) Limiting distributions for branching random fields. Trans. Amer. Math. Soc. 239, 353–389.Google Scholar
Gel'Fand, I. M. and Vilenkin, N. Ya. (1964) Generalized Functions, Vols. I and IV. Academic Press, New York.Google Scholar
Ikeda, N., Nagasawa, M. and Watanabe, S. (1968–69) Branching Markov processes, I, II, III. J. Math. Kyoto Univ. 8, 233–278, 363–410; 9, 95–160.Google Scholar
Ivanoff, B. G. (1976) Branching Diffusion Processes. Ph.D. Thesis, Carleton University, Ottawa.Google Scholar
Jagers, P. (1974) Aspects of random measures and point processes. Advances in Probability 3, ed. Ney, P., Dekker, New York, 179–238.Google Scholar
Jones, D. S. (1966) Generalized Functions. McGraw-Hill, New York.Google Scholar
Moyal, J. E. (1957) Discontinuous Markov processes. Acta Math. 98, 221–264.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google Scholar
Moyal, J. E. (1964) Multiplicative population processes. J. Appl. Prob. 1, 267–283.Google Scholar
Sawyer, S. (1970) A formula for semi-groups, with an application to branching diffusion processes. Trans. Amer. Math. Soc. 152, 1–38.Google Scholar
Sawyer, S. (1975) An application of branching random fields to genetics. In Probability Methods in Differential Equations, Springer Lecture Notes in Maths 451, Springer-Verlag, Berlin, 100–112.Google Scholar
Sawyer, S. (1976) Branching diffusion processes in population genetics. Adv. Appl. Prob. 8, 659–689.Google Scholar
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.Google Scholar