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The branching diffusion with immigration

Published online by Cambridge University Press:  14 July 2016

B. G. Ivanoff
B. G. Ivanoff*
Affiliation:
Université de Montréal
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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 

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