Abstract
We prove local limit theorems for the total mass processes of two branching- uctuating particle systems which converge to discontinuous (2; d; fl)-superprocess. To this end, we establish new subtle properties of the total mass for this class of superpro- cesses. Thus, the density of its absolutely continuous component exhibits a polynomial blow-up at the origin and has a regularly varying upper tail. Both particle systems consid- ered are characterized by the same heavy-tailed branching mechanism that belongs to the domain of normal attraction of an extreme stable law with index 1 + fl 2 (1; 2). One of them starts from a Poisson eld, whereas the initial number of particles for the other sys- tem is non-random. We demonstrate that the poissonization of the initial eld of particles is related to Gnedenko's method of accompanying innitely divisible laws. The compar- ison of our results with their 'continuous' counterparts (which pertain to convergence to the super-Brownian motion) reveals a worse discrepancy between the extinction proba- bilities. This is explained through the intrinsic difference between structures of individual surviving clusters.
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