  We define a characteristic function for probability measures on the
signatures of geometric rough paths. We determine sufficient conditions under
which a random variable is uniquely determined by its expected signature, thus
partially solving the analogue of the moment problem. We furthermore study
analyticity properties of the characteristic function and prove a method of
moments for weak convergence of random variables. We apply our results to
signature arising from L\'evy, Gaussian and Markovian rough paths.
