  We investigate coarsening and persistence in the voter model by introducing
the quantity $P_n(t)$, defined as the fraction of voters who changed their
opinion n times up to time t. We show that $P_n(t)$ exhibits scaling behavior
that strongly depends on the dimension as well as on the initial opinion
concentrations. Exact results are obtained for the average number of opinion
changes, <n>, and the autocorrelation function, $A(t)\equiv \sum (-1)^n P_n\sim
t^{-d/2}$ in arbitrary dimension d. These exact results are complemented by a
mean-field theory, heuristic arguments and numerical simulations. For
dimensions d>2, the system does not coarsen, and the opinion changes follow a
nearly Poissonian distribution, in agreement with mean-field theory. For
dimensions d<=2, the distribution is given by a different scaling form, which
is characterized by nontrivial scaling exponents. For unequal opinion
concentrations, an unusual situation occurs where different scaling functions
correspond to the majority and the minority, as well as for even and odd n.
