Number variance for hierarchical random walks and related fluctuations

TL;DR

This paper investigates the variance and fluctuation behavior of hierarchical random walks, establishing conditions for bounded variance, and demonstrating Gaussian fluctuation limits with Markovian properties, contrasting with stable process counterparts.

Contribution

It provides necessary and sufficient conditions for variance boundedness and characterizes the Gaussian fluctuation limits for hierarchical random walks, highlighting differences from stable process models.

Findings

Variance remains bounded under specific conditions related to transience/recurrence.

Fluctuations converge to a Gaussian process with Markovian properties.

Hierarchical random walks exhibit different fluctuation structures compared to stable processes.

Abstract

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the c-random walks. Such walks are used, e.g., in population genetics. The number variance problem consists in investigating if the variance of the number of "particles" N_n(L) lying in the ball of radius L at a given time n remains bounded, or even better, converges to a finite limit, as $L\to \infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of N_n(L) around the mean as $n\to \infty$ and L is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the c-random walks mimic symmetric stable processes on R, we compare our results to those obtained by Hambly and Jones (2007,2009), where the number variance problem for an infinite system of symmetric stable processes on R was studied. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a centered Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments.