Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric α-stable processes

Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric α-stable processes are used to construct explicit 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.