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We show that the centred occupation time process of the origin of a system ofcritical binary branching random walks in dimension $d\ge 3$, started offeither from a Poisson field or in equilibrium, when suitably normalized,converges to a Brownian motion in $d\ge4$. In $d=3$, the limit process is afractional Brownian motion with Hurst parameter 3/4 when starting inequilibrium, and a related Gaussian process when starting from a Poisson field.For (dependent) branching random walks with state dependent branching rate weobtain convergence in f.d.d. to the same limit process, and for $d=3$ also afunctional limit theorem.

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