Spatial Central Limit Theorem for Supercritical Superprocesses

Piotr Miłoś ()

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 1-40

Abstract: Abstract We consider a measure-valued diffusion (i.e., a superprocess). It is determined by a couple $$(L,\psi )$$ ( L , ψ ) , where L is the infinitesimal generator of a strongly recurrent diffusion in $$\mathbb {R}^{d}$$ R d and $$\psi $$ ψ is a branching mechanism assumed to be supercritical. Such processes are known, see for example, (Englander and Winter in Ann Inst Henri Poincaré 42(2):171–185, 2006), to fulfill a law of large numbers for the spatial distribution of the mass. In this paper, we prove the corresponding central limit theorem. The limit and the CLT normalization fall into three qualitatively different classes arising from “competition” of the local growth induced by branching and global smoothing due to the strong recurrence of L. We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total mass of the process.

Keywords: Branching processes; Supercritical branching processes; Limit behavior; Central limit theorem; Primary 60F05; 60J80; Secondary 60G20 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-016-0704-6

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