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 Lévy, Gaussian and Markovian rough paths.
"Characteristic functions of measures on geometric rough paths." Ann. Probab. 44 (6) 4049 - 4082, November 2016. https://doi.org/10.1214/15-AOP1068