A functional CLT for the occupation time of a state-dependent branching random walk

Abstract

We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d\ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in $d\ge 4$. In $d=3$, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for $d=3$ also a functional limit theorem.

Published in at http://dx.doi.org/10.1214/009117907000000150 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• A long range dependence stable process and an infinite variance branching system
• Occupation time fluctuations of an infinite variance branching system in large dimensions
• Occupation time fluctuations of Poisson and equilibrium finite variance branching systems

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• Particle picture interpretation of some Gaussian processes related to fractional Brownian motion
• Occupation times of subcritical branching immigration systems with Markov motion, clt and deviations principles