# Number variance for hierarchical random walks and related fluctuations - Mathematics > Probability

Abstract: We study an infinite system of independent symmetric random walks on ahierarchical group, in particular, the c-random walks. Such walks are used,e.g., in population genetics. The number variance problem consists ininvestigating if the variance of the number of -particles- N nL lying in theball of radius L at a given time n remains bounded, or even better, convergesto a finite limit, as $L\to \infty$. We give a necessary and sufficientcondition and discuss its relationship to transience-recurrence property of thewalk. Next we consider normalized fluctuations of N nL around the mean as$n\to \infty$ and L is increased in an appropriate way. We prove convergence offinite dimensional distributions to a Gaussian process whose properties arediscussed. As the c-random walks mimic symmetric stable processes on R, wecompare our results to those obtained by Hambly and Jones 2007,2009, wherethe number variance problem for an infinite system of symmetric stableprocesses on R was studied. Since the hierarchical group is an ultrametricspace, corresponding results for symmetric stable processes and hierarchicalrandom walks may be analogous or quite different, as has been observed in othercontexts. An example of a difference in the present context is that for thestable processes a fluctuation limit process is a centered Gaussian processwhich is not Markovian and has long range dependent stationary increments, butthe counterpart for hierarchical random walks is Markovian, and in a specialcase it has independent increments.

Author: Tomasz Bojdecki, Luis G. Gorostiza, Anna Talarczyk

Source: https://arxiv.org/