Asymptotic fluctuations and critical dimension for a two-level branching system

Abstract

The high-density asymptotic behaviour of a two-level branching system in R^d is studied. In the finite-variance case, a fluctuation limit process is obtained which is characterized as a generalized Ornstein-Uhlenbeck process. In the case of critical branching at the two levels the long-time behaviour of the fluctuation limit process is shown to have critical dimension 2α, where α is the index of the symmetric stable process representing the underlying particle motion. The same critical dimension has been obtained recently for the related (but qualitatively different) two-level superprocess. The fluctuation analysis uses different and simpler tools than the superprocess analysis.