  We obtain Central Limit Theorems in Functional form for a class of
time-inhomogeneous interacting random walks on the simplex of probability
measures over a finite set. Due to a reinforcement mechanism, the increments of
the walks are correlated, forcing their convergence to the same, possibly
random, limit. Random walks of this form have been introduced in the context of
urn models and in stochastic approximation. We also propose an application to
opinion dynamics in a random network evolving via preferential attachment. We
study, in particular, random walks interacting through a mean-field rule and
compare the rate they converge to their limit with the rate of synchronization,
i.e. the rate at which their mutual distances converge to zero. Under certain
conditions, synchronization is faster than convergence.
