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

   
@article{EJP419,
	author = {Ben Hambly and Liza Jones},
	title = {Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {12},
	year = {2007},
	keywords = {Number variance; symmetric alpha- stable processes; controlled variability; Gaussian fluctuations;  functional limits;  long memory; Gaussian processes; fractional Brownian motion},
	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.},
	pages = {no. 30, 862-887},
	issn = {1083-6489},
	doi = {10.1214/EJP.v12-419},    
        url = {http://ejp.ejpecp.org/article/view/419}}