  Some probabilistic aspects of the number variance statistic are investigated.
Infinite systems of independent Brownian motions and symmetric alpha-stable
processes are used to construct new examples of processes which exhibit both
divergent and saturating number variance behaviour. We derive a general
expression for the number variance for the spatial particle configurations
arising from these systems and this enables us to deduce various limiting
distribution results for the fluctuations of the associated counting functions.
In particular, knowledge of the number variance allows us to introduce and
characterize a novel family of centered, long memory Gaussian processes. We
obtain fractional Brownian motion as a weak limit of these constructed
processes.
