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2007 Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.
Ben Hambly, Liza Jones
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Electron. J. Probab. 12: 862-887 (2007). DOI: 10.1214/EJP.v12-419

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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Ben Hambly. Liza Jones. "Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.." Electron. J. Probab. 12 862 - 887, 2007. https://doi.org/10.1214/EJP.v12-419

Information

Accepted: 13 June 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60046
MathSciNet: MR2318413
Digital Object Identifier: 10.1214/EJP.v12-419

Subjects:
Primary: 60G15 , 60G52
Secondary: 15A52 , 60F17

Keywords: controlled variability , fractional Brownian motion , functional limits , Gaussian fluctuations , Gaussian processes , long memory , Number variance , symmetric alpha- stable processes

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Vol.12 • 2007
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