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1 results

  • The branching random field

  • Gail Ivanoff
  • Journal:
    Advances in Applied Probability / Volume 12 / Issue 4 / December 1980
    Published online by Cambridge University Press:
    01 July 2016, pp. 825-847
    Print publication:
    December 1980

  • The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.