Fluctuations of the occupation times for branching system starting from infinitely divisible point processes

Abstract

In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to α-stable motion in R^d, α<d<2α. The particles split according to the binary critical branching law with intensity V>0. We study how the limit behaviour of the fluctuations of the occupation time depends on the \emph{initial particle configuration}. We obtain a functional central limit theorem for a vast class of infinitely divisible distributions. Our findings extend and put in a unified setting results which previously seemed to be disconnected. The limit processes form a one dimensional family of long-range dependance centred Gaussian processes.

Proofs shortened, extended discussion

References in corpus (3)

• Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses
• Occupation times of subcritical branching immigration system with Markov motions
• Occupation times of subcritical branching immigration systems with Markov motion, clt and deviations principles

Cited by in corpus (1)

• Particle systems with quasi-homogeneous initial states and their occupation time fluctuations