Electronic Journal of Probability

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

Ben Hambly and Liza Jones

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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.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 30, 862-887.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G52: Stable processes 60G15: Gaussian processes
Secondary: 60F17: Functional limit theorems; invariance principles 15A52

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

This work is licensed under aCreative Commons Attribution 3.0 License.


Hambly, Ben; Jones, Liza. Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes. Electron. J. Probab. 12 (2007), paper no. 30, 862--887. doi:10.1214/EJP.v12-419. https://projecteuclid.org/euclid.ejp/1464818501

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