  Occupation time fluctuation limits of particle systems in R^d with
independent motions (symmetric stable Levy process, with or without critical
branching) have been studied assuming initial distributions given by Poisson
random measures (homogeneous and some inhomogeneous cases). In this paper, with
d=1 for simplicity, we extend previous results to a wide class of initial
measures obeying a quasi-homogeneity property, which includes as special cases
homogeneous Poisson measures and many deterministic measures (simple example:
one atom at each point of Z), by means of a new unified approach. In previous
papers, in the homogeneous Poisson case, for the branching system in "low"
dimensions, the limit was characterized by a long-range dependent Gaussian
process called sub-fractional Brownian motion (sub-fBm), and this effect was
attributed to the branching because it had appeared only in that case. An
unexpected finding in this paper is that sub-fBm is more prevalent than
previously thought. Namely, it is a natural ingredient of the limit process in
the non-branching case (for "low" dimension), as well. On the other hand,
fractional Brownian motion is not only related to systems in equilibrium (e.g.,
non-branching system with initial homogeneous Poisson measure), but it also
appears here for a wider class of initial measures of quasi-homogeneous type.
