The Annals of Probability

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

Matthias Birkner and Iljana Zähle

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We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension d≥3, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in d≥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.

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Ann. Probab., Volume 35, Number 6 (2007), 2063-2090.

First available in Project Euclid: 8 October 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

State dependent branching random walk occupation time functional central limit theorem


Birkner, Matthias; Zähle, Iljana. A functional CLT for the occupation time of a state-dependent branching random walk. Ann. Probab. 35 (2007), no. 6, 2063--2090. doi:10.1214/009117907000000150.

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