## Electronic Journal of Probability

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

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1464818501

Digital Object Identifier
doi:10.1214/EJP.v12-419

Mathematical Reviews number (MathSciNet)
MR2318413

Zentralblatt MATH identifier
1127.60046

Rights

#### Citation

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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