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Supercritical Branching Random Fields. Asymptotics of a Process Involving the Past

  • Luis G. Gorostiza
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 9)

Abstract

We consider a branching random field with immigration on Rd, described as follows. Initial particles appear at time t = 0 according to a Poisson random field with intensity measure γdx, γ ≧ 0, on a Borel set B⊂Rd. Immigrant particles appear according to a Poisson random field with intensity measure ßdxdt, ß ≧ 0, on a Borel set C⊂Rd × [0,∞). The two Poisson fields are independent. As time evolves, the particles independently migrate following standard Brownian motions during independent exponentially distributed lifetimes with parameter V, at the end of which they independently reproduce with a branching law {pn} n=0 (i.e. a particle branches into n particles with probability pn) having finite second moment; the mean and the second factorial moment of the branching law will be denoted m1 and m2 respectively. The offspring particles appear at the locations where their parent particles branched, and independently migrate, live and reproduce by the same laws. A more general model will be mentioned later on.

Keywords

Standard Brownian Motion Factorial Moment Functional Central Limit Theorem Descendance Line Infinite Particle 
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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Luis G. Gorostiza
    • 1
  1. 1.Centro de Investigación y de Estudios AvanzadosMéxico D.F.México

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