Limiting behaviour of a signature

Signature of a path provides a top down summary of the path as a driving signal. There have been substantial recent progress in reconstructing paths from its signature, (Lyons-Xu 2016, Geng 2016). In this talk, we focus on obtaining certain quantitative features of paths from their signatures. Hambly-Lyons' showed that the normalized limit of signature gives the length of a C^3 path. The result was recently extended by Lyons-Xu to C^1 paths. The extension of this result to bounded variation paths remains open. We will discuss this open problem.