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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 30 open journal systems 


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.


Full text: PDF

Pages: 862-887

Published on: June 13, 2007


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Electronic Journal of Probability. ISSN: 1083-6489