The branching random field
- 1 December 1980
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 12 (4) , 825-847
- https://doi.org/10.2307/1426743
Abstract
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.Keywords
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