Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length
Jiawei Chang, Terry Lyons, Hao Ni · Comptes Rendus Mathématique · 2018
For a path of length L > 0 , if for all n ≥ 1 , we multiply the n -th term of the signature by n ! L − n , we say that the resulting signature is ‘ normalised ’. It has been established (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the n -th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the n -th root of the norm of the n -th term in the normalised signature as n approaches infinity.