Signature asymptotics, empirical processes, and optimal transport

Rough path theory [15] provides one with the notion of the signature, a graded family of tensors which characterise, up to a negligible equivalence class, an ordered stream of vector-valued data. In this article, we lay down the theoretical foundations for a connection between signature asymptotics, the theory of empirical processes, and Wasserstein distances, opening up the landscape and toolkit of the second and third in the study of the first. Our main contribution is to show that the Hilbert-Schmidt norm of the signature can be reinterpreted as a statement about the asymptotic behaviour of Wasserstein distances between two independent empirical measures of samples from the same underlying distribution. In the setting studied here, these measures are derived from samples from a probability distribution which is directly determined by geometrical properties of the underlying path. The general question of rates of convergence for these objects has been studied in depth in the recent monograph of Bobkov and Ledoux [2]. To illustrate this new connection, we show how the above main result can be used to prove a more general version of the original asymptotic theorem of Hambly and Lyons [19]. We conclude by providing an explicit way to compute that limit in terms of a second-order differential equation.