Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.
Ben M Hambly, University of Oxford
Liza A Jones, University of Oxford
Abstract
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 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.
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