  We study the limit fluctuations of the rescaled occupation time process of a
branching particle system in $\mathbb{R}^d$, where the particles are subject to
symmetric $\alpha$-stable migration ($0<\alpha\leq2$), critical binary
branching, and general non-lattice lifetime distribution. We focus on two
different regimes: lifetime distributions having finite expectation, and
Pareto-type lifetime distributions, i.e. distributions belonging to the normal
domain of attraction of a $\gamma$-stable law with $\gamma\in(0,1)$.
  In the latter case we show that, for dimensions
$\alpha\gamma<d<\alpha(1+\gamma)$, the rescaled occupation time fluctuations
converge weakly to a centered Gaussian process whose covariance function is
explicitly calculated, and we call it {\em weighted sub-fractional Brownian
motion.} Moreover, in the case of lifetimes with finite mean, we show that for
$\alpha<d<2\alpha$ the fluctuation limit turns out to be the same as in the
case of exponentially distributed lifetimes studied by Bojdecki et al. [7,8,9].
We also investigate the maximal parameter range allowing existence of the
weighted sub-fractional Brownian motion and provide some of its fundamental
properties, such as path continuity, long-range dependence, self-similarity and
the lack of Markov property.
