  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 $d\ge4$. 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.
