  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.
